Hydrologic Design Storms

Reading: Applied Hydrology Sections 14-1 to 14-4

2

Hydrologic design

• Water control– Peak flows, erosion, pollution, etc.

• Water management– Domestic and industrial use, irrigation, instream flows, etc

• Tasks– Determine design inflow– Route the design inflow– Find the output

• check if it is sufficient to meet the demands (for management)• Check if the outflow is at safe level (for control)

3

4

Return Period

• Random variable:• Threshold level:• Extreme event occurs if: • Recurrence interval: • Return Period:

Average recurrence interval between events equalling or exceeding a threshold

• If p is the probability of occurrence of an extreme event, then

or

TxX

Tx

X

TxX of ocurrencesbetween Time

)(E

pTE

1)(

TxXP T

1)(

5

More on return period

• If p is probability of success, then (1-p) is the probability of failure

• Find probability that (X ≥ xT) at least once in N years.

NN

T

TT

T

T

TpyearsNinonceleastatxXP

yearsNallxXPyearsNinonceleastatxXP

pxXP

xXPp

111)1(1)(

)(1)(

)1()(

)(

6

Risk Analysis

• Uncertainty in hydrology – Inherent - stochastic nature of hydrologic phenomena– Model – approximations in equations– Parameter – estimation of coefficients in equations

• Consideration of Risk– Structure may fail if event exceeds T–year design

magnitude

– R = P(event occurs at least once in n years)• Natural inherent risk of failure

nTxXPR )(11 T

xXP T

1)(

n

TR

111

7

Example 13.2.2• Expected life of culvert = 10 yrs• Acceptable risk of 10 % for the culvert

capacity• Find the design return period

yrsT

T

TR

n

95

11110.0

111

10

What is the chance that the culvert designed for an event of 95 yr return period will have its capacity exceeded at least once in 50 yrs?

41.0

95

111

50

R

R

The chance that the capacity will not be exceeded during the next 50 yrs is 1-0.41 = 0.59

8

Hydroeconomic Analysis

• Probability distribution of hydrologic event and damage associated with its occurrence are known

• As the design period increases, capital cost increases, but the cost associated with expected damages decreases.

• In hydroeconomic analysis, find return period that has minimum total (capital + damage) cost.

9

Design Storms

• Get Depth, Duration, Frequency Data for the required location

• Select a return period• Convert Depth-Duration data to a design

hyetograph.

Depth Duration Data to Rainfall Hyetograph

12

TP 40

• Hershfield (1961) developed isohyetal maps of design rainfall and published in TP 40.

• TP 40 – U. S. Weather Bureau technical paper no. 40. Also called precipitation frequency atlas maps or precipitation atlas of the United States.– 30mins to 24hr maps for T = 1 to 100

• Web resources for TP 40 and rainfall frequency maps– http://www.tucson.ars.ag.gov/agwa/rainfall_frequency.ht

ml– http://www.erh.noaa.gov/er/hq/Tp40s.htm– http://hdsc.nws.noaa.gov/hdsc/pfds/

13

2yr-60min precipitation GIS map

14

2yr-60min precipitation map

This map is from HYDRO 35 (another publication from NWS) which supersedes TP 40

http://hdsc.nws.noaa.gov/hdsc/pfds/index.html

An example of precipitation frequency estimates for a location in California

37.4349 N120.6062 W

Results of Precip Frequency Query

18

Design areal precipitation

• Point precipitation estimates are extended to develop an average precipitation depth over an area

• Depth-area-duration analysis – Prepare isohyetal maps from point precipitation

for different durations– Determine area contained within each isohyet– Plot average precipitation depth vs. area for each

duration

Tropical Storm Allison

TS Allison

21

Depth-area curve

(World Meteorological Organization, 1983)

TS Allison

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 170

1

2

3

4

5

6

Time (Hours)

Rain

fall

(inch

es)

0 2 4 6 8 10 12 14 16 18 200

5

10

15

20

25

30

Cumulative Rainfall

Extreme Rainfalls, Station 1600

0 2 4 6 8 10 12 14 16 18 200

5

10

15

20

25

30

Cumulative Rainfall

Time Rainfall 2-Hour 3-Hour 4-hour1.00 0.002.00 0.04 0.043.00 1.18 1.22 1.224.00 0.98 2.16 2.20 2.205.00 2.48 3.46 4.64 4.686.00 2.40 4.88 5.86 7.047.00 1.46 3.86 6.34 7.328.00 1.69 3.15 5.55 8.039.00 4.88 6.57 8.03 10.43

10.00 3.39 8.27 9.96 11.4211.00 3.86 7.25 12.13 13.8212.00 3.19 7.05 10.44 15.3213.00 0.00 3.19 7.05 10.4414.00 0.20 0.20 3.39 7.2515.00 0.08 0.28 0.28 3.4716.00 0.04 0.12 0.32 0.3217.00 0.00 0.04 0.12 0.32

Depth 4.88 8.27 12.13 15.32Intensity 4.88 4.13 4.03 3.83

Values in inches for depth and inches per hour for intensity

24

Depth (intensity)-duration-frequency

• DDF/IDF – graph of depth (intensity) versus duration for different frequencies– TP 40 or HYDRO 35 gives spatial distribution of

rainfall depths for a given duration and frequency– DDF/IDF curve gives depths for different durations

and frequencies at a particular location– TP 40 or HYDRO 35 can be used to develop

DDF/IDF curves

• Depth (P) = intensity (i) x duration (Td) diTP

25

IDF curve

26

Example 14.2.1

• Determine i and P for a 20-min duration storm with 5-yr return period in Chicago

From the IDF curve for Chicago,

i = 3.5 in/hr for Td = 20 min and T = 5yr

P = i x Td = 3.5 x 20/60 = 1.17 in

27

Equations for IDF curves

IDF curves can also be expressed as equations to avoid reading from graphs

fT

ci

ed

i is precipitation intensity, Td is the duration, and c, e, f are coefficients that vary for locations and different return periods

fT

cTi

ed

m

This equation includes return period (T) and has an extra coefficient

(m)

28

Example 14.2.4

Using IDF curve equation, determine 10-yr 20-min design rainfall intensities for Denver

fT

ci

ed

From Table 14.2.3 in the text, c = 96.6, e = 0.97, and f = 13.9

hrini /002.39.1320

6.9697.0

Similarly, i = 4.158 and 2.357 in/hr for Td = 10 and 30 min, respectively

29

IDF curves for Austin

cbt

ai

tscoefficien,,

stormofDuration

intensityrainfalldesign

cba

t

i

Storm Frequency a b c

2-year 106.29 16.81 0.9076

5-year 99.75 16.74 0.8327

10-year 96.84 15.88 0.7952

25-year 111.07 17.23 0.7815

50-year 119.51 17.32 0.7705

100-year 129.03 17.83 0.7625

500-year 160.57 19.64 0.7449

0

2

4

6

8

10

12

14

16

1 10 100 1000

Duration (min)

Inte

nsi

ty (

in/h

r)

2-yr

5-yr

10-yr

25-yr

50-yr

100-yr

500-yr

Source: City of Austin, Watershed Management Division

30

Design Precipitation Hyetographs

• Most often hydrologists are interested in precipitation hyetographs and not just the peak estimates.

• Techniques for developing design precipitation hyetographs

1. SCS method2. Triangular hyetograph method3. Using IDF relationships (Alternating block method)

TS Allison

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 170

1

2

3

4

5

6

Time (Hours)

Rain

fall

(inch

es)

32

SCS MethodSCS (1973) adopted method similar to DDF to develop dimensionless rainfall temporal patterns called type curves for four different regions in the US.SCS type curves are in the form of percentage mass (cumulative) curves based on 24-hr rainfall of the desired frequency.If a single precipitation depth of desired frequency is known, the SCS type curve is rescaled (multiplied by the known number) to get the time distribution. For durations less than 24 hr, the steepest part of the type curve for required duraction is used

33

SCS type curves for Texas (II&III)

SCS 24-Hour Rainfall Distributions SCS 24-Hour Rainfall Distributions

T (hrs) Fraction of 24-hr rainfall T (hrs) Fraction of 24-hr rainfall

Type II Type III Type II Type III

0.0 0.000 0.000 11.5 0.283 0.298

1.0 0.011 0.010 11.8 0.357 0.339

2.0 0.022 0.020 12.0 0.663 0.500

3.0 0.034 0.031 12.5 0.735 0.702

4.0 0.048 0.043 13.0 0.772 0.751

5.0 0.063 0.057 13.5 0.799 0.785

6.0 0.080 0.072 14.0 0.820 0.811

7.0 0.098 0.089 15.0 0.854 0.854

8.0 0.120 0.115 16.0 0.880 0.886

8.5 0.133 0.130 17.0 0.903 0.910

9.0 0.147 0.148 18.0 0.922 0.928

9.5 0.163 0.167 19.0 0.938 0.943

9.8 0.172 0.178 20.0 0.952 0.957

10.0 0.181 0.189 21.0 0.964 0.969

10.5 0.204 0.216 22.0 0.976 0.981

11.0 0.235 0.250 23.0 0.988 0.991

24.0 1.000 1.000

Choice of Calibration Storms

Representative?– In temporal shape

2007 Storm 2010 Hermine Storm

35

SCS Method Steps

• Given Td and frequency/T, find the design hyetograph

1. Compute P/i (from DDF/IDF curves or equations)2. Pick a SCS type curve based on the location 3. If Td = 24 hour, multiply (rescale) the type curve with P to

get the design mass curve1. If Td is less than 24 hr, pick the steepest part of the type curve

for rescaling

4. Get the incremental precipitation from the rescaled mass curve to develop the design hyetograph

36

Example – SCS Method• Find - rainfall hyetograph for a 25-year, 24-hour duration SCS

Type-III storm in Harris County using a one-hour time increment

• a = 81, b = 7.7, c = 0.724 (from Tx-DOT hydraulic manual)

• Find – Cumulative fraction - interpolate SCS table– Cumulative rainfall = product of cumulative fraction * total 24-hour

rainfall (10.01 in)– Incremental rainfall = difference between current and preceding

cumulative rainfall

hrin

bt

ai c /417.0

7.760*24

81724.0

inhrhrinTiP d 01.1024*/417.0*

TxDOT hydraulic manual is available at: http://manuals.dot.state.tx.us/docs/colbridg/forms/hyd.pdf

37

SCS – Example (Cont.)Time Cumulative

Fraction Cumulative Precipitation

Incremental Precipitation

(hours) Pt/P24 Pt (in) (in)

0 0.000 0.00 0.00 1 0.010 0.10 0.10 2 0.020 0.20 0.10 3 0.032 0.32 0.12 4 0.043 0.43 0.12 5 0.058 0.58 0.15 6 0.072 0.72 0.15 7 0.089 0.89 0.17 8 0.115 1.15 0.26 9 0.148 1.48 0.33

10 0.189 1.89 0.41 11 0.250 2.50 0.61 12 0.500 5.01 2.50 13 0.751 7.52 2.51 14 0.811 8.12 0.60 15 0.849 8.49 0.38 16 0.886 8.87 0.38 17 0.904 9.05 0.18 18 0.922 9.22 0.18 19 0.939 9.40 0.18 20 0.957 9.58 0.18 21 0.968 9.69 0.11 22 0.979 9.79 0.11 23 0.989 9.90 0.11 24 1.000 10.01 0.11

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Time (hours)

Pre

cip

itat

ion

(in

)

If a hyetograph for less than 24 needs to be prepared, pick time intervals that include the steepest part of the type curve (to capture peak rainfall). For 3-hr pick 11 to 13, 6-hr pick 9 to 14 and so on.

38

Triangular Hyetograph Method

• Given Td and frequency/T, find the design hyetograph1. Compute P/i (from DDF/IDF curves or equations)2. Use above equations to get ta, tb, Td and h (r is available for

various locations)

Time

Rain

fall

inte

nsity

, i

h

ta tb

d

a

T

tr

Td

Td: hyetograph base length = precipitation duration

ta: time before the peak

r: storm advancement coefficient = ta/Td

tb: recession time = Td – ta = (1-r)Td

d

d

T

Ph

hTP

22

1

39

Triangular hyetograph - example

• Find - rainfall hyetograph for a 25-year, 6-hour duration in Harris County. Use storm advancement coefficient of 0.5.

• a = 81, b = 7.7, c = 0.724 (from Tx-DOT hydraulic manual)

hrin

bt

ai

c/12.1

7.760*6

81724.0

inhrhriniP 72.66*/12.16*

hrtTt

hrrTt

adb

da

336

365.0

Time

Rain

fall

inte

nsity

, in/

hr

2.24

3 hr 3 hr

6 hr

hrinT

Ph

d

/24.26

44.13

6

72.622

40

Alternating block method• Given Td and T/frequency, develop a hyetograph in

Dt increments1. Using T, find i for Dt, 2Dt, 3Dt,…nDt using the IDF curve

for the specified location2. Using i compute P for Dt, 2Dt, 3Dt,…nDt. This gives

cumulative P.3. Compute incremental precipitation from cumulative P.4. Pick the highest incremental precipitation (maximum

block) and place it in the middle of the hyetograph. Pick the second highest block and place it to the right of the maximum block, pick the third highest block and place it to the left of the maximum block, pick the fourth highest block and place it to the right of the maximum block (after second block), and so on until the last block.

41

Cumulative Incremental Duration Intensity Depth Depth Time Precip (min) (in/hr) (in) (in) (min) (in) 10 4.158 0.693 0.693 0-10 0.024 20 3.002 1.001 0.308 10-20 0.033 30 2.357 1.178 0.178 20-30 0.050 40 1.943 1.296 0.117 30-40 0.084 50 1.655 1.379 0.084 40-50 0.178 60 1.443 1.443 0.063 50-60 0.693 70 1.279 1.492 0.050 60-70 0.308 80 1.149 1.533 0.040 70-80 0.117 90 1.044 1.566 0.033 80-90 0.063 100 0.956 1.594 0.028 90-100 0.040 110 0.883 1.618 0.024 100-110 0.028 120 0.820 1.639 0.021 110-120 0.021

Example: Alternating Block Method

90.13

6.9697.0

d

ed TfT

ci

tscoefficien,,

stormofDuration

intensityrainfalldesign

fec

T

i

d

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80 80-90 90-100

100-110

110-120

Time (min)

Pre

cip

itat

ion

(in

)

Find: Design precipitation hyetograph for a 2-hour storm (in 10 minute increments) in Denver with a 10-year return period 10-minute