Surface Water Hydrology
• Summarized in one
equation
Q  VA
V = velocity, fps or m/s
A = channel cross-sectional area, sf or m2
Stream
Discharge (Q)
• Fundamental stream
variable
• Cubic feet/second
(cfs, ft-3sec-1)
• Q=V*A
– Velocity (ft/sec) *
– Channel area (ft2)
• Instantaneous
A
V
Streamflow Measurement
Figure 11.30
Cut Away of a Gage House
Stilling Well, Static Tube, Staff
Gage
USGS Midsection Method
Q = ΣQi where
Qi = (ui * di) * {( bi+1 – bi-1)/2}
For d 4 use
(b5-b3)/2
http://www.caf.wvu.edu/~forage/streamflow/estimat.htm
Measure velocity at .6 x depth, as measured from the water surface
For deep channels, measure velocity at 0.2 x depth and 0.8 x depth, from WS
Stage and Discharge Graph for Gallatin River at Gallatin
Gateway, MT 2001-2002
4000
3500
Discharge (cfs)
3000
2500
2000
1500
1000
500
0
0
0.5
1
1.5
2
2.5
Stage (ft)
3
3.5
4
4.5
Case Study:
Yellowstone
River at
Livingston, MT
http://water.usgs.gov/waterwatch/
• Manually
measured
occasionally to
define rating
curve: Q=VA
• Continuously
monitored: stage
Yellowstone
River II
• Synthesized for
floods /
droughts
• Summarized for
risk /
geomorphic
work
• http://waterwatch.usgs.gov/
Flood flow prediction using
statistical methods
Probability distribution functions of
flood flow events
• Theoretical distributions
– Gumbel
– Log Pearson type III (used in “Guidelines for
calculation of flood flow frequency”, Water
Resources Council Bulletin 17B)
– Weibull
the distribution of the maximum of a number of samples
Gumbel distribution
 e b
F ( x)  e
b 1
x  x  0.45s
.7797s
x  flood magnit ude



x  averageflood magnit ude
s  st andarddeviat ionof flood magnit udes
• Used to describe annual flow events
• Fits an extreme value distribution to flood
data by assuming the annual flood is the
largest of a sample of 365 possible values
per year
• Exceedence probability = 1-F(x)
Gumbel distribution:
Blacksmith Fork nr. Hyrum, UT
100
90
80
70
return period
60
50
40
30
20
10
0
0
200
400
600
800
1000
annual peak flood magnitude
1200
1400
1600
1800
Log Pearson type III distribution
 
logQr .i .   log Q  K r .i . logs 
Qr .i .  floodflow of desired recurrenceint erval
Q  averageof flow series
s  st andarddeviat ionof flow series
K r .i .  t ypeIII deviat e
• Exceedence probabilities are estimated using a normal
distribution with adjustments for skewness of the flow
series
• Flow values are transformed to logarithmic values
• Average and standard deviation of the transformed
values, and estimates of the skewness of the distribution
are used to estimate the flood flow of a given recurrence
interval
• K value = f(skew value, desired recurrence interval)
• Skew value = f(number of years of record)
Log Pearson type III distribution
peak
669
124
1020
1620
289
880
975
1100
998
415
362
264
738
842
467
244
• Step 1: Transform flow
data to log domain
• Step 2: Calculate mean
and standard deviation of
log-transformed data set
• Step 3: Calculate
weighted skew
– G = station skew
– G-bar = generalized skew
(from Plate 1 of Bull. 17B)
– MSE = mean square error
of each skew (from pgs.
13-14 of Bull. 17B)
Gw 
G
log(peak)
2.825426
2.093422
3.0086
3.209515
2.460898
2.944483
2.989005
3.041393
2.999131
2.618048
2.558709
2.421604
2.868056
2.925312
2.669317
2.38739
Descriptive Statistics
Mean
Standard Error
Median
Mode
Standard Deviation
Sample Variance
Kurtosis
Skewness
Range
Minimum
Maximum
Sum
Count
2.635905
0.033384
2.668851
2.060698
0.309594
0.095848
-0.68429
-0.30834
1.192178
2.025306
3.217484
226.6878
86
 
MSEG G   MSEG G
MSEG  MSEG

N X  X

3
N  1N  2 s 3
N  number in dat a set
X  log of flow value
X  mean of all log flow values
s  st andarddeviat ionof all flow values
Log Pearson type III distribution
• Step 4: Using weighted skew and return
interval of interest (e.g. 2 yr flood), look up
type III deviate, K, in Appendix 3, Bulletin
17B
• Calculate log of flow of desired return
interval
• E.g. at Blacksmith Fork nr. Hyrum, UT
– log(Q2)=mean(log(Q’s))+K*standard deviation(log(Q’s))
– log(Q2)=2.64+(0.06)*(0.31)
– Q2 = 455 cfs
Weibull distribution
m
n  1
• Also called “plotting position”, p
• Estimated cumulative probabilities
– implies the largest observed value is the max possible
value of the distribution
• Sort n data from smallest to largest, assign a
rank m, to the sorted data, and use m and n to
calculate an empirical cumulative probability
• Recurrence interval
– 1/p
Weibull Distribution
Discharge Value
10,000
6,200
5,000
500
m
Rank , m
1
2
3
50
n
Recurrence interval
(n+1/m)
Q
Return Interval
Weibull distribution:
Blacksmith Fork nr. Hyrum, UT
100
90
80
Return period (yrs)
70
60
50
40
30
20
10
0
0
200
400
600
800
1000
Annual peak flood magnitude (cfs)
1200
1400
1600
1800
A comparison of Gumbel, log
Pearson type III, and Weibull
distributions: calculation of Q2
Gumbel
Log Pearson
type III
Weibull
484 cfs
451 cfs
466.5 cfs
What if there isn’t a gaging
station?
• Predict discharge from physiographic
variables
– drainage area, annual precip. (inches), %
forested area,
• Q = f(A)
• SW Montana:
– Q2 = 2.48 A0.87 (HE+10)0.19
• Omang, 92; USGS WRIR 92-4048
Linear Regression
• Method of determining correlation between 2 (or more) variables
• The quantity r, called the linear correlation coefficient, measures the
strength and the direction of a linear relationship between two
variables
• The coefficient of determination, r 2, is useful because it gives the
proportion of the variance (fluctuation) of one variable that is
predictable from the other variable
– It is a measure that allows us to determine how certain one can be in
making predictions from a certain model/graph
• The coefficient of determination is the ratio of the explained variation
to the total variation
• For example, if r = 0.922, then r 2 = 0.850, which means that 85% of
the total variation in y can be explained by the linear relationship
between x and y (as described by the regression equation). The
other 15% of the total variation in y remains unexplained.
Table 2.6
Annual Precip. for LA, CA 1934-1953
Year Depth (in) Rank Depth (in) Year
1934
14.6
1
32.8 1941
1935
21.7
2
26.2 1952
1936
12.1
3
23.4 1938
1937
22.4 You
4 have determined
22.4 1937 that
1938
23.4
5
21.7 1935
23.4”1940
of annual
1939
13.1 more
6 than 19.2
1940
19.2 rainfall
7
19.2
1944in a net
will
result
1941
32.8
8
18.2 1943
loss
your
1942
11.2 Economic
9
14.6 for
1934
1943
18.2 crop.
10
13.1 1939
1944
19.2
11
12.7 1947
1945
11.6 Now,
12 you need
12.1 to
1936
predict
1946
11.6
13
11.6 1945
How
often
this 1946
will occur.
1947
12.7
14
11.6
1948
7.2
15
11.2 1942
1949
8
16
10.6 1950
1950
10.6
17
9.5 1953
1951
8.2
18
8.2 1951
1952
26.2
19
8 1949
1953
9.5
20
7.2 1948
Table 2.6
Annual Precip. for LA, CA 1934-1953
Year Depth (in) Rank Depth (in) Year
1934
14.6
1
32.8 1941
1935
21.7
2
26.2 1952
1936
12.1
3
23.4 1938
1937
22.4
4
22.4 1937
1938
23.4
5
21.7 1935
1939
13.1
6
19.2 1940
1940
19.2
7
19.2 1944
1941
32.8
8
18.2 1943
1942
11.2
9
14.6 1934
1943
18.2
10
13.1 1939
1944
19.2
11
12.7 1947
1945
11.6
12
12.1 1936
1946
11.6
13
11.6 1945
1947
12.7
14
11.6 1946
1948
7.2
15
11.2 1942
1949
8
16
10.6 1950
1950
10.6
17
9.5 1953
1951
8.2
18
8.2 1951
1952
26.2
19
8 1949
1953
9.5
20
7.2 1948
Weibull
Probability of Occurrence, Fa (%)=
100 *
n
y+1
n = the rank of each event
y = the total number of events
For Example:
Year 1938--23.4 in--Rank #3
100 (3)
20
= 15.0 %
Hazen
Probability of Occurrence, Fa (%)=
100 (2n-1)
2y
n = the rank of each event
y = the total number of events
For Example:
Year 1938--23.4 in--Rank #3
100 (2*3-1)
2*20
= 12.5%
Annual Pre cip. for LA, CA 1934-1953
Ye ar
De pth (in)
Rank
De pth (in)
Ye ar
Prob. F
1934
14.6
1
32.8
1941
2.5
1935
21.7
2
26.2
1952
7.5
1936
12.1
3
23.4
1938
12.5
1937
22.4
4
22.4
1937
17.5
1938
23.4
5
21.7
1935
22.5
1939
13.1
6
19.2
1940
27.5
1940
19.2
7
19.2
1944
32.5
1941
32.8
8
18.2
1943
37.5
1942
11.2
9
14.6
1934
42.5
1943
18.2
10
13.1
1939
47.5
1944
19.2
11
12.7
1947
52.5
1945
11.6
12
12.1
1936
57.5
1946
11.6
13
11.6
1945
62.5
1947
12.7
14
11.6
1946
67.5
1948
7.2
15
11.2
1942
72.5
1949
8
16
10.6
1950
77.5
1950
10.6
17
9.5
1953
82.5
1951
8.2
18
8.2
1951
87.5
a
Return Period=
100
Fa
Fa= probability of occurrence (%)
For Example:
Year 1938-- Fa= 12.5
100 = 8 yrs OR 100/15 = 6.7 yrs
12.5
Table 2.6
Annual Precip. for LA, CA 1934-1953
Year Depth (in) Rank Depth (in) Year Prob. F
1934
14.6
1
32.8 1941
2.5
1935
21.7
2
26.2 1952
7.5
1936
12.1
3
23.4 1938
12.5
1937
22.4
4
22.4 1937
17.5
1938
23.4
5
21.7 1935
22.5
1939
13.1
6
19.2 1940
27.5
1940
19.2
7
19.2 1944
32.5
1941
32.8
8
18.2 1943
37.5
1942
11.2
9
14.6 1934
42.5
1943
18.2
10
13.1 1939
47.5
1944
19.2
11
12.7 1947
52.5
1945
11.6
12
12.1 1936
57.5
1946
11.6
13
11.6 1945
62.5
1947
12.7
14
11.6 1946
67.5
1948
7.2
15
11.2 1942
72.5
1949
8
16
10.6 1950
77.5
1950
10.6
17
9.5 1953
82.5
1951
8.2
18
8.2 1951
87.5
1952
26.2
19
8 1949
92.5
1953
9.5
20
7.2 1948
97.5
a
Return Period
40.0
13.3
8.0
5.7
4.4
3.6
3.1
2.7
2.4
2.1
1.9
1.7
1.6
1.5
1.4
1.3
1.2
1.1
1.1
1.0
Frequency-Magnitude Graph
Peak Streamflow for the Nation
USGS 06043500 Gallatin River near Gallatin Gateway MT
Yr
Date
Gage Ht (ft) Discharge (cfs)
1995
Jun. 06, 1995
6.31
7,330
1996
Jun. 11, 1996
6.11
7,010
1997
Jun. 02, 1997
6.71
9,160
1998
Jun. 26, 1998
5.18
5,400
1999
May 29, 1999
4.98
4,990
2000
May 29, 2000
4.44
4,070
2001
May 15, 2001
4.26
3,740
2002
Jun. 02, 2002
5.03
5,260
2003
May 30, 2003
5.71
6,710
2004
Jun. 10, 2004
4.30
3,810
2005
Jun. 23, 2005
4.52
4,220
2006
May 21, 2006
5.17
5,550
2007
May 14, 2007
4.04
3,350
2008
Jun. 23, 2008
5.54
6,010
2009
Jun. 01, 2009
5.28
5,500
Fa(%) = 100* (n/y+1)
Recurrence Interval (Return Period) = 100/Fa
Using the entire 82 yrs of record,
R.I.(9160) = 82 yrs
R.I.(5500) = 2.4 yrs
What’s the recurrence interval of the largest flood of the last 15 yrs?
What’s the recurrence interval of the 2009 runoff peak?
16 yrs
2.7 yrs
Other types of flood predictions:
fish migration flows
–
•
•
•
•
•
Estimation of Migration Flows during High Flow Periods
The magnitude of migration flows during high flow periods was estimated
for the Upper Main and South Fork Red River watersheds using the 10
percent exceedance flow determined for the April to June period for each
year of available data.
The April to June period was based on migration timing of steelhead trout
and westslope cutthroat trout.
Mean daily flow data for April, May, and June for each year of record was
sorted from largest to smallest. A rank was then assigned to the sorted
data to calculate an empirical cumulative probability.
The high flow for migration was estimated as the flow at which the
probability that a given flow will be equaled or exceeded is 10 percent.
A second estimate of high flows for migration was determined as 50
percent of the 2-year flood flow. The 2-year flood, as determined using
the Log Pearson Type III distribution, was used to estimate migration
flows during high flow periods.
Flow Duration Curve for S. Fk. Red River
(April – Jun for the period of record)
600
Mean Daily Discharge (cfs)
500
400
300
200
100
0
0.00
10.00
20.00
30.00
40.00
50.00
60.00
Percent of Time Exceeded
70.00
80.00
90.00
100.00
Estimation of Migration Flows
during Low Flow Periods
•
•
•
•
•
•
The magnitude of migration flows during low flow periods was estimated
for the Upper Main and South Fork Red River watersheds using the 95
percent exceedance flow determined using data for August and
September.
The August through September timing was based on spawning timing of
chinook salmon and bull trout in the Newsome Creek watershed.
Mean daily flow data for August and September for each year of record
was sorted from largest to smallest.
A rank was then assigned to the sorted data to calculate an empirical
cumulative probability.
The low flow for migration was estimated as the flow at which the
probability that a given flow will be equaled or exceeded of 95 percent of
the time.
A second estimate of low flows for migration was determined by using the
7-day average for mean daily flow in August and September. For each
year of record, a 7-day average was calculated for flows in August and
September. The minimum 7-day average for each year was selected as
the 7-day average for that year.
Flow Duration Curve for S. Fk. Red River
(Aug-Sept for the period of record)
50
45
Mean Daily Discharge (cfs)
40
35
30
25
20
15
10
5
0
0.00
10.00
20.00
30.00
40.00
50.00
60.00
Percent of Time Exceeded
70.00
80.00
90.00
100.00