04/20/2006
Design Storms
Reading: Applied Hydrology Sec 14.1 – 14.4
Design Storm
•
•
•
Design storm – precipitation pattern defined
for use in the design of hydrologic system
Serves as an input to the hydrologic system
Can by defined by:
1. Hyetograph (time distribution of rainfall)
2. Isohyetal map (spatial distribution of rainfall)
2
Extreme value (EV) distributions
• Extreme values – maximum or minimum
values of sets of data
• Annual maximum discharge, annual minimum
discharge
• When the number of selected extreme values
is large, the distribution converges to one of
the three forms of EV distributions called Type
I, II and III
3
EV type I distribution
• If M1, M2…, Mn be a set of daily rainfall or streamflow,
and let X = max(Mi) be the maximum for the year. If
Mi are independent and identically distributed, then
for large n, X has an extreme value type I or Gumbel
distribution.
f ( x) 

 x u
 x  u 
exp 
 exp 


  
 
1
6sx

u  x  0.5772
Distribution of annual maximum streamflow follows an EV1 distribution
4
EV type III distribution
• If Wi are the minimum streamflows
in different days of the year, let X =
min(Wi) be the smallest. X can be
described by the EV type III or
Weibull distribution.
 k  x 
f ( x)    
    
k 1
  x k 
exp   
    
x  0;  , k  0
Distribution of low flows (eg. 7-day min flow)
follows EV3 distribution.
5
Design point precipitation
• Historic data of precipitation is available
• Precipitation data are converted to different
durations (Table 3.4.1)
• Annual maximum precipitation for a given
duration is selected for each year
• Frequency analysis is performed to derive
design precipitation depths for different
return periods
• The depths are converted to intensities by
dividing by precipitation durations
6
IDF curves by frequency analysis
1. Get annual maximum series of precipitation
depth for a given duration
2. Use EV1/Gumbel distribution to find
precipitation depth for different return
periods
3. Repeat 1 and 2 process for different
durations
4. Plot depth versus duration for different
frequencies
7
IDF curve
8
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
9
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/
10
24-hour Design Rainfall Totals
http://onlinemanuals.txdot.gov/txdotmanuals/hyd/ebdlkup.xls
Rainfall Frequency Analysis from TP-40
tc = time of concentration in minutes (not less than 10 minutes)
I = rainfall intensity (inches/hour)
http://onlinemanuals.txdot.gov/txdotmanuals/hyd/the_rational_method.htm#i999837
Rainfall Frequency Analysis in Texas
I
b
I
(t c  d ) e
2642
I
(tc  8.8) 0.805
2642
I
(1440 8.8) 0.805
For tc = 24 hours = 24*60 = 1440 min, I = 7.53 inches/hour
2yr-60min precipitation map
This map is from
HYDRO 35 (another
publication from
NWS) which
supersedes TP 40
14
Design precipitation for Austin
15
IDF curves for Austin
i
i  design rainfall intensity
t  Duration of storm
a
t  b c
a, b, c  coefficien ts
Storm Frequency
a
b
c
16
2-year
106.29
16.81
0.9076
14
5-year
99.75
16.74
0.8327
2-yr
5-yr
10-yr
25-yr
50-yr
100-yr
500-yr
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
Intensity (in/hr)
12
10
8
6
4
2
0
1
500-year
160.57
19.64
0.7449
Source: City of Austin, Watershed Management Division
10
100
1000
Duration (min)
16
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 method
2. Triangular hyetograph method
3. Using IDF relationships (Alternating block method)
17
SCS Method
SCS
(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
18
SCS type curves for Texas (II&III)
SCS 24-Hour Rainfall Distributions
T (hrs)
SCS 24-Hour Rainfall Distributions
Fraction of 24-hr rainfall
Type II
T (hrs)
Type III
Fraction of 24-hr rainfall
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
19
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 curve
1.
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
20
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)
i
a
81

 0.417in / hr
t  bc 24* 60  7.70.724
P  i *Td  0.417in / hr * 24 hr  10.01in
• 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
TxDOT hydraulic manual is available at:
http://manuals.dot.state.tx.us/docs/colbridg/forms/hyd.pdf
21
SCS – Example (Cont.)
(hours)
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
Cumulative
Fraction
Cumulative
Precipitation
Incremental
Precipitation
Pt/P24
Pt (in)
(in)
0.000
0.010
0.020
0.032
0.043
0.058
0.072
0.089
0.115
0.148
0.189
0.250
0.500
0.751
0.811
0.849
0.886
0.904
0.922
0.939
0.957
0.968
0.979
0.989
1.000
0.00
0.10
0.20
0.32
0.43
0.58
0.72
0.89
1.15
1.48
1.89
2.50
5.01
7.52
8.12
8.49
8.87
9.05
9.22
9.40
9.58
9.69
9.79
9.90
10.01
0.00
0.10
0.10
0.12
0.12
0.15
0.15
0.17
0.26
0.33
0.41
0.61
2.50
2.51
0.60
0.38
0.38
0.18
0.18
0.18
0.18
0.11
0.11
0.11
0.11
3.00
2.50
Precipitation (in)
Time
2.00
1.50
1.00
0.50
0.00
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)
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.
22
Triangular Hyetograph Method
Rainfall intensity, i
ta
Td: hyetograph base length = precipitation duration
tb
ta: time before the peak
r
ta
Td
h
Td
Time
r: storm advancement coefficient = ta/Td
tb: recession time = Td – ta = (1-r)Td
1
P  Td h
2
2P
h
Td
• Given Td and frequency/T, find the design hyetograph
1. 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)
23
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)
a
81

 1.12in / hr
c
t  b 6 * 60  7.70.724
h
2 P 2  6.72 13.44


 2.24 in / hr
Td
6
6
t a  rTd  0.5  6  3 hr
tb  Td  ta  6  3  3 hr
P  i * 6  1.12in / hr * 6 hr  6.72 in
Rainfall intensity, in/hr
i
24
3 hr
3 hr
2.24
6 hr
Time
Alternating block method
• Given Td and T/frequency, develop a hyetograph in
Dt increments
1. Using T, find i for Dt, 2Dt, 3Dt,…nDt using the IDF curve for
the specified location
2. 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.
25
Example: Alternating Block Method
Find: Design precipitation hyetograph for a 2-hour storm (in 10
minute increments) in Denver with a 10-year return period 10minute
Duration
(min)
10
20
30
40
50
60
70
80
90
100
110
120
Td 
e
Intensity
(in/hr)
4.158
3.002
2.357
1.943
1.655
1.443
1.279
1.149
1.044
0.956
0.883
0.820
f

Td 
Cumulative
Depth
(in)
0.693
1.001
1.178
1.296
1.379
1.443
1.492
1.533
1.566
1.594
1.618
1.639
i  design rainfall intensity
96 .6
0.97
 13 .90
Incremental
Depth
(in)
0.693
0.308
0.178
0.117
0.084
0.063
0.050
0.040
0.033
0.028
0.024
0.021
Td  Duration of storm
c, e, f  coefficien ts
0.8
Time
(min)
0-10
10-20
20-30
30-40
40-50
50-60
60-70
70-80
80-90
90-100
100-110
110-120
Precip
(in)
0.024
0.033
0.050
0.084
0.178
0.693
0.308
0.117
0.063
0.040
0.028
0.021
0.7
0.6
Precipitation (in)
i
c
0.5
0.4
0.3
0.2
0.1
0.0
0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80 80-90
26
Time (min)
90100
100110
110120
Design aerial 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
27
Depth-area curve
(World Meteorological28Organization, 1983)
Study by Will Asquith, USGS
http://pubs.usgs.gov/wri/wri99-4267/pdf/wri99-4267.pdf
http://pubs.usgs.gov/wri/wri99-4267/pdf/wri99-4267.pdf
http://pubs.usgs.gov/wri/wri99-4267/pdf/wri99-4267.pdf
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)
P  iTd
32
Probable Maximum Precipitation
•
Probable maximum precipitation
– Greatest depth of precipitation for a given duration that
is physically possible and reasonably characteristic over a
particular geographic region at a certain time of year
– Not completely reliable; probability of occurrence is
unknown
•
Variety of methods to estimate PMP
1. Application of storm models
2. Maximization of actual storms
3. Generalized PMP charts
33
Probable Maximum Storm
• Probable maximum storm
– Temporal distribution of rainfall
– Given as maximum accumulated depths for a
specified duration
– Information on spatial and temporal distribution
of PMP is required to develop probable maximum
storm hyetograph
34
Probable Maximum Flood
• PMF – greatest flood to be expected assuming
complete coincidence of all factors that would
produce the heaviest rainfall (PMP) and maximum
runoff
– Flood of unknown frequency
– Most structures are not designed for PMF, but for greatest
floods that may be reasonably expected for local
conditions (meteorology, topography, and hydrology)
– The design flood is commonly called standard project flood
derived from standard project storm
35