UNIT HYDROGRAPH THEORY
Input, I Output, OTransferFunction
Ot = f{It,It-1,It-1, ......, It-n}
RainfallHyetograph
DrainageBasin
DischargeHydrograph
Time
Rai
nfal
l
0
10
20
TimeD
isch
arge
0
10
20 IO
LUMPED HYDROLOGIC MODELS
Before GEO3280!
BLACK BOX
UNIT HYDROGRAPH THEORY
Sherman - 1932Horton - 1933Wisler & Brater - 1949 - “the hydrograph of
surface runoff resulting from a relatively short, intense rain, called a unit storm.”
The runoff hydrograph may be “made up” of runoff that is generated as flow through the soil (Black, 1990).
WHAT IS THE THEORY BEHIND THE UNIT HYDROGRAPH ?
Impervious Parking Lot
OutletContributing A7A6A5A4A3A2A1Time
A1PA1P0-1A2PA2P1-2A3PA3P2-3A4PA4P3-4A5PA5P4-5A6PA6P5-6A7PA7P6-7
A unit of precipitation, P, falling for 1 minute, uniformlyover the entire area.
1mm (0.001m) of precipitation falling for 1 minute, uniformlyover the entire area.
OutletVolumeContributing Areas (m2)
(m3)17032835042443024454Time0.0540.0540-10.2440.2441-20.4300.4302-30.4240.4243-40.3500.3504-50.3280.3285-60.1700.1706-7
Time (mins)
1 2 3 4 5 6 7 8 9 10 11
1 2 3 4 5 6 7 8 9 10 11
Dis
char
ge a
t Out
let (
m3 m
in-1
)
0.0
0.2
0.4
0.6
0.8
1.0
Prec
ipita
tion
(mm
)
01234
56
Outflow Hydrograph in Response to 1mm (0.001m)of Precipitation Falling for 1 minute, Uniformly
over the Entire Area.
OutletVolumeContributing Areas (m2)
(m3)17032835042443024454Time0.0540.0540-10.2440.2441-20.4300.4302-30.4240.4243-40.3500.3504-50.3280.3285-60.1700.1706-7
ASSUMPTIONS:
1. RESPONSE IS ALWAYS DIRECTLY PROPORTIONAL TO THE INPUT.
Time (mins)
1 2 3 4 5 6 7 8 9 10 11
1 2 3 4 5 6 7 8 9 10 11
Prec
ipita
tion
(mm
)
Dis
char
ge a
t Out
let (
m3 m
in-1
)
0.0
0.2
0.4
0.6
0.8
1.0 01234
56
Outflow Hydrograph in Response to 2mm (0.002m)of Precipitation Falling for 1 minute, Uniformly
over the Entire Area.
OutletVolumeContributing Areas (m2)
(m3)17032835042443024454Time0.1080.1080-10.4880.4881-20.8600.8602-30.8480.8483-40.7000.7004-50.6560.6565-60.3400.3406-7
NOTE:
RESPONSE IS ALWAYS PROPORTIONATE TO THE INPUT. DOUBLE INPUT- DOUBLE RESPONSE.
Time (mins)
1 2 3 4 5 6 7 8 9 10 11
1 2 3 4 5 6 7 8 9 10 11
Dis
char
ge a
t Out
let (
m3 m
in-1
)
0.0
0.2
0.4
0.6
0.8
1.0
Prec
ipita
tion
(mm
)01234
56
Outflow Hydrograph in Response to 2.2mm (0.0022m)of Precipitation Falling for 1 minute, Uniformly
over the Entire Area.
OutletVolumeContributing Areas (m2)
(m3)17032835042443024454Time0.1190.1190-10.5370.5371-20.9460.9462-30.9330.9333-40.7700.7704-50.7220.7225-60.3740.3746-7
NOTE:
TWO POINT TWO TIMES THE INPUT- TWO POINT TWO TIMES THE RESPONSE!
ASSUMPTIONS:
1. RESPONSE IS ALWAYS DIRECTLY PROPORTIONAL TO THE INPUT.
2. TIME-BASE OF RESPONSE IS ALWAYSFIXED REGARDLESS OF THE INPUT
Time (mins)
1 2 3 4 5 6 7 8 9 10 11
1 2 3 4 5 6 7 8 9 10 11
Dis
char
ge a
t Out
let (
m3 m
in-1
)
0.0
0.2
0.4
0.6
0.8
1.0 01234
56
Outflow Hydrograph in Response to 1mm (0.002m)of Precipitation Falling for 2 Consecutive Minutes, Uniformly
over the Entire Area.
VolumeContributing Areas (m2)(m3)17032835042443024454Time
0.0540.0540-10.2980.2440.0541-20.6740.4300.2442-30.8540.4240.4303-40.7740.3500.4244-50.6780.3280.3505-60.4980.1700.3286-70.1700.1707-8
NOTE:
TWO CONSECUTIVE INPUTSOF 1mm/min PRODUCESTWO CONSECUTIVE, BUTSUPERIMPOSED RESPONSES,THE TIME BASES OF WHICHARE EQUAL.
Outflow Hydrograph in Response to 0.25mm, 2.1mm and 0.8mmof Precipitation Falling for 3 Consecutive Minutes, Uniformly
over the Entire Area.
VolumeContributing Areas (m2)(m3)17032835042443024454Time
0.0140.0140-10.1740.0610.1131-20.6630.1080.5120.0432-31.2040.1060.9030.1953-41.3220.0880.8900.3444-51.1560.0820.7350.3395-61.0110.0430.6890.2806-70.6190.3570.2627-80.1360.1368-9
Time (minutes)
0 1 2 3 4 5 6 7 8 9 10
Pre
cip
(mm
)
Dis
char
ge a
t Out
let (
m3 m
in-1
)
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.401
2
3
NOTE:
THE TIME BASE OF EACHRESPONSE REMAINS AT 7MINUTES, REGARDLESS OF WHETHER THE IMPUT IS 0.22 OR 1 mm/minutes AS IN THIS MORE COMPLEX EVENT
HOW CAN A UNIT HYDROGRAPHBE DERIVED IN REALITY?
Days since April 30, 1969
0 30 60 90 120 150 180 210 240 270 300 330 360
Mea
n D
aily
Dis
char
ge (m
3 s-1
)
0
20
40
60
80
100
120
140
May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr
DAILY DISCHARGE AT ELECTRIONA
1. Select fairly clearly defined single events from available records
Avoid this sort of complex response where it is hard to separate one storm from the next
Days since April 30, 1969.
205 210 215 220 225
Mea
n D
aily
Dis
char
ge (m
3 s-1
)
0
20
40
60
80
100
120
140Dec. 1Nov 25 Dec. 13
2. Extract observed hydrograph
Days since April 30, 1969.
205 210 215 220 225
Mea
n D
aily
Dis
char
ge (m
3 s-1
)
0
20
40
60
80
100
120
140Dec. 1Nov 25 Dec. 13
3. Separate out the “Direct Runoff” response from the groundwater or baseflow.
GROUNDWATER?
Direct RunoffHydrograph
1. Define point at which flood hydrograph rises2. Superimpose Master Recession curve of declining limb.3. Identify “Inflection Point” at which baseflow is dominant process4. Join Time of Rise to Inflection Point with straight line5. Graphically separate Direct Runoff Hydrograph from Base Flow
HYDROGRAPH SEPARATION
Time (T)
0 10 20 30 40 50
Sta
ge (L
)
0
20
40
60
80
100
120
140
160
Time ofRise
InflectionPoint
MasterRecession
Straight Line
DRH
BASE FLOW
Days since April 30, 1969
0 30 60 90 120 150 180 210 240 270 300 330 360
Mea
n D
aily
Dis
char
ge (m
3 s-1
)
0
20
40
60
80
100
120
140
May Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr
DAILY DISCHARGE AT ELECTRIONAHOW TO COMPUTE MASTER RECESSION CURVE
CREATING MASTER RECESSION CURVE
Time (T)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Sta
ge (L
)
15
20
25
30
35
40
Master Recession
IndividualRecession
Curves
4. Determine the quantity and temporal distribution of “Effective Rainfall” that gave rise to that Direct Runoff Hydrograph
As is assignments and lectures, this involves consideration of water lost to interception, depression storage and infiltration.
Average StormEffective Precipitation, p
Storm Duration, d
Effective Hydrograph37.6mm over 90 minutes
Time (mins)0 60 120 180 240 300 360 420 480 540 600 660 720 780
Run
off (
mm
)
0
20
40
60
80
100
120
5. Determine the Direct Runoff Hydrograph resulting from an Effective Precipitation of p mm over d hours/minutes in duration
Direct RunoffHydrograph
Effective Hyetograph
HOW DO YOU PROCEED FROM THIS 37.6 mm 90 min UNIT HYDROGRAPH TO ONE OF MORE CONVENTIONAL
DEPTH (e.g. 25mm) AND TIME (e.g. 1 hour) UNITS?
1. Use the Equal Time Base of Response assumption
Time (minutes)
0 60 120 180 240 300 360 420 480 540 600 660 720 780
0 60 120 180 240 300 360 420 480 540 600 660 720 780R
unof
f (m
m)
0
50
100
150
200
250
300 0
2040
Precip.
(mm
)
90mins
90mins
• This S-curve showing the response to a very large number of sequential 37.6mmm in 90 events would be the same no matter what time we actually started the sequence, at time 0, or after 5 minutes, or 60 minutes.
• Imagine that we set such a sequence running at time 0, and a second identical sequence running at time 60 minutes.
• We have two s-curves offset by 60 minutes.• The difference in discharges between the two S-
curves, measured at any time, represents the response of the first S-curve, that the second one hasn’t yet experienced because of the 60 minute shift.
• This difference therefore represents the “missing” rainfall in that 60 minute period, or the 37.6mm for 60 minutes hydrograph.
Time (mins)0 60 120 180 240 300 360 420 480 540 600 660 720 780
0 60 120 180 240 300 360 420 480 540 600 660 720 780R
unof
f (m
m)
0
50
100
150
200
250
0
50
100
150
200
250
OFFSET 37.6mm/90min S-CURVE BY 60 mins.
Time (mins)
0 60 120 180 240 300 360 420 480 540 600 660 720 780
Run
off (
mm
)
0
20
40
60
800 60 120 180 240 300 360 420 480 540 600 660 720 780
0 60 120 180 240 300 360 420 480 540 600 660 720 780
Run
off (
mm
)
0
50
100
150
200
250
0
50
100
150
200
250
Offset 37.6mm-90 min
S-Curves
37.6mm-60 min UnitHydrograph
HOW DO WE CONVERT THIS 37.6mm for 1hr UNIT HYDROGRAPH INTO
SOMETHING MORE STANDARD LIKE A 25mm (1”) for 1 hour UNIT
HYDROGRAPH?
Use the assumption of a proportional response.
Time (mins)0 60 120 180 240 300 360 420 480 540 600 660 720 780
0 60 120 180 240 300 360 420 480 540 600 660 720 780
Run
off (
mm
)
0
20
40
60
80
0
20
40
60
80
Derive 25.4mm-60min U.H.Multiply x-coordinate
by (25.4/37.6)
37.6mm - 60 min U.H.25.4 mm(1")-1hr U.H.
DERIVED 25.4mm (1")-1 hr. UNIT HYDROGRAPH
Time (mins)
0 60 120 180 240 300 360 420 480 540 600 660
0 60 120 180 240 300 360 420 480 540 600 660
Run
off (
mm
)
0
10
20
30
40
50
60
0
10
20
30
40
50
60 HERE IS OUR BASIC “BUILDING BLOCK”, DERIVED WITHOUT HAVING TO GO THROUGH THE EXERCISE OF ISOCHRONES ETC., BUT WHICH CAN BE USED TO FORECAST RESPONSES TO STORM DELIVERING VARYING QUANTITIES OF WATER IN DIVERSE PATTERNS OVER TIME.