Analyzing Hydrologic Time-Series for Flood Frequency Analysis
Analyzing Hydrologic Time-Series
a road map
Flood Frequency Analysis
Hydrologic Engineering Center
Beth Faber, PhD, PE
Lecture 1.5
Goals
Review (or preview) some ways we might study a time-series,
and what we can learn from it
1.
What information we extract
2.
What assumptions we make
3.
What values or probabilities we can estimate
4.
What type of analysis each requires
2
3
Topics
•
Annual Extremes
–
annual maximums or minimums, Bulletin 17B/C
–
instantaneous flows or longer duration avg flow/volume
•
Partial Duration (peaks over threshold)
•
Daily flows or stages
–
Duration Curves
–
Summary Hydrographs
•
Annual Volumes
–
Drought, Dependent random variables
No independence - persistence
independence
3
Assumption
independence
No independence - persistence
4
Topics
•
Annual Extremes
–
annual maximums
or minimums,
Bulletin 17B/C
–
instantaneous flows
or longer duration avg flow/volume
•
Partial Duration (peaks over threshold)
•
Daily flows or stages
–
Duration Curves
–
Summary Hydrographs
•
Annual Volumes
–
Drought, Dependent random variables
4
No independence - persistence
independence
Assumption
independence
No independence - persistence
Streamflow data, daily flow record
American River at Fair Oaks (unregulated)
5
reservoir
constructed
often, annual peaks are
instantaneous
values
American River at Fair Oaks (unregulated)
6
Streamflow data, find annual max
American River, ANNUAL MAX SERIES, AMS
annual maximum values are independent
7
7
Assumptions
•
What we have:
series of annual maximum flows
•
Treat these flows as a random,
representative sample
from the population of interest
•
Generally, we
assume
the sample is
IID
–
annual peak flows are random and
independent
–
peak flows are
identically distributed
– homogeneous data set
–
sample is adequately representative of the population
(but estimate of the distribution improves with sample size)
NOTE, since we collect data over time, we’re assuming the
data/watershed is stationary….
8
-- probability distribution of annual
peak flow
divide data range into bins, count number of observations
in each bin (
frequency
), divide by total (
relative frequency
)
9
Histogram
this shape is similar to a PDF, probability density function
0 25 50 75 100 125 150 175 200 225 250 275
9
10
Histogram
3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0 5.2 5.4 5.6
switch
to
log
10
(flow)
divide data range into bins, count number of observations
in each bin (
frequency
), divide by total (
relative frequency
)
10
this shape is similar to a PDF, probability density function
L
o
g
F
l
o
w
(
t
h
o
u
s
a
n
d
s
o
f
C
F
S
)
Definition of PDF and CDF
A
Probability Density Function (PDF)
, f(x),
defines the probability of occurrence for
a continuous random variable.
area under curve = probability
The
Cumulative Distribution Function
(CDF)
, F(x) is the probability the random
variable is less than some value
curve = probability
11
p
p
X
X
11
CDF
o
f
V
a
r
i
a
b
l
e
X
o
f
V
a
r
i
a
b
l
e
X
12
Empirical Cumulative Distribution
Empirical
or Graphical estimate: based on
order statistics
, using
plotting position
(estimated non-exceedance probability)
–
Cumulative probability of observation magnitude is based on its
position within the sample
, and the
sample size
–
Each observation has incremental prob = 1/N
m
i
= rank of ordered observation i
(m=1 as smallest value, N as largest)
N = total # of observations
estimate probability that outcome will be smaller than observation x
i
by the fraction of the observed sample that
is
smaller than x
i
based on observation
i.e.,
relative
frequency
12
Plotting Positions
Want to avoid a value equal to 1 (N/N)!
Some common plotting positions:
Weibull
mean estimate of probability
Median
median estimate of probability
Hirsch-Stedinger
based on threshold-exceedance
where
m = rank
N = total # of values
Empirical Cumulative Distribution
Plotting Position:
General Formula
Weibull: a = 0.0, b = 1.0
Median: a = -0.3, b = 0.4
plotting position
P =
m
+ a
N
+ b
14
15
CDF = cumulative
distribution function
Empirical Cumulative Distribution
15
Empirical Cumulative Distribution
16
for a Corps
frequency curve,
we switch axes...
16
annual
17
0
1
...and adjust axis scales
Normal probability scale
Empirical Cumulative Distribution
…also called a frequency curve…
17
annual
18
log
scale
0
1
...and adjust axis scales
Normal probability scale
Empirical Cumulative Distribution
…also called a frequency curve…
18
annual
19
Cumulative
Exceedance
Probability
1
0
log
scale
equaled or
exceeded 11
times in 108
years
≈
10%
switch to
exceedance
Normal probability scale
Empirical Cumulative Distribution
…also called a frequency curve…
19
Annual
Weibull would
be 1/109
20
Cumulative
Exceedance
Probability
1
0
Normal probability scale
log
scale
Empirical
Graphical
Cumulative Distribution
20
Annual
21
Cumulative
Exceedance
Probability
1
0
Normal probability scale
log
scale
Log Pearson III per
Bulletin 17B/C
for unregulated,
instantaneous annual
maximums
21
Annual
Ana
lytical
Distribution
Distribution Fitting – a model for probability:
Step 1: Choose Model
Step 2: Calibrate Model (Estimate Distribution Parameters)
Can estimate population parameters using sample statistics.
The general
descriptive parameters
are:
•
Central Tendency
where?
(location)
•
Dispersion or Spread
how wide?
(scale)
•
Asymmetry
(shape)
symmetrical?
Distribution
Moments
22
23
dimensionless
where: X
i
= sample member, i
N
= sample size
Sample Mean
Sample Standard
Deviation
Sample Skew
Coefficient
S
= 0
> 0
< 0
average distance
from mean
Distribution Moments
24
Distribution Parameters
P
e
a
k
CDF, frequency curve form
25
100 x probability of exceeding
changing
the skew
P
e
a
k
100 x probability of exceeding
CDF, frequency curve form
Distribution Parameters
skew = 0 is
straight line
on this
Normal prob
axis
26
27
Cumulative
Exceedance
Probability
1
0
Normal probability scale
log
scale
Log Pearson III per
Bulletin 17B/C
for unregulated,
instantaneous annual
maximums
27
Annual
Ana
lytical
Distribution
Bulletin
17B
dated March
1982
Bulletin
17C
dated Feb
2018
28
28
•
Estimate
LogPearson III
distribution for unregulated annual peak flows
–
Method of Moments
•
Data challenges:
–
Missing Flows (Broken or Incomplete Record)
–
Low and High Outliers
–
Zero Flows
•
Additional information to improve estimates:
–
Historical/Paleo Information
–
Regional Skews (use a
weighted
skew)
–
Allowing a longer record site to improve short record estimate
B17B/C: Frequency Analysis of Annual Peak Flows
moments of sample used to estimate
moments of distribution
many of the methods
for handling data
challenges and using
additional information
are improved in
Bulletin 17C
29
29
30
Cumulative
Exceedance
Probability
1
0
Normal probability scale
log
scale
90%
confidence
interval
Log Pearson III per
Bulletin 17B/C
30
Annual
for unregulated,
instantaneous annual
maximums
Ana
lytical
Distribution
Also Interested in
Regulated
Flow
31
Reservoir Storage Level
Inflow
Release
115,000 cfs
160,000 cfs
31
Streamflow data, regulated
32
Is this regulation homogeneous?
period of record 1955 - 2009
homogeneous
=
identically-
distributed
=
stationary
32
reservoir
constructed
Streamflow data, regulated
33
these are simulated outflows:
HOMOGENEOUS operation
period of record 1921 - 2002
2 benefits:
homogeneous
operation,
and longer
period
33
Streamflow data, regulated
regulated annual peaks
34
34
Folsom Release, ANNUAL MAX
35
annual maximum values are independent
35
36
Plotted Annual Maximums
Cumulative Exceedance Probability
1
0
36
37
Does an Analytical Curve Make Sense?
Cumulative Exceedance Probability
1
0
37
38
Graphical Curve is Better for this
Cumulative Exceedance Probability
1
0
38
39
Graphical Curve is Better for this
Cumulative Exceedance Probability
how do we
extrapolate?
39
Extending a Regulated Curve
•
To extend a regulated frequency curve, we can
construct synthetic events
with a specified frequency,
e.g. 100-yr, 200-yr, etc
–
Estimate frequency curves for longer duration average flows
or volumes
(3-day, 7-day, 30-day, etc)
–
Construct hydrograph for a given frequency
–
Route through operation model to get peak outflow
–
NOTE: will assume likelihood of peak release is related to
likelihood of peak inflow
40
40
41
Graphical Curve is Better for this
200
500
Cumulative Exceedance Probability
41
42
Topics
•
Annual Extremes
–
annual maximums or minimums, Bulletin 17B/C
–
Instantaneous flows or
longer duration avg flow/volume
•
Partial Duration (peaks over threshold)
•
Daily flows or stages
–
Duration Curves
–
Summary Hydrographs
•
Annual Volumes
–
Drought, Dependent random variables
No independence - persistence
independence
42
Assumption
independence
No independence - persistence
43
Longer Durations of Flow/Volume
Volume
Frequency Analysis
Create a family of frequency curves for a given site that
specify
average flow
across various durations
1.
Compute m-day moving average TS
for various durations
, m
•
m might be 1-day, 5-day, 30-day etc.
2.
Extract annual maximums
for each duration
3.
Perform frequency analysis
for each duration
•
Similar to Bulletin 17B/C procedure, but not instantaneous peak
•
Compute plotting positions and distribution parameters from each
period of record sample
43
Longer Durations Max Average Flows
44
44
Longer Duration Max Average Flows
45
45
Annual Maximums of Various Durations
46
h
i
g
h
e
s
t
3
-
d
a
y
a
v
e
r
a
g
e
46
47
Max Flow at Folsom Volume-Freq Curves
Percent Chance Exceedance
Average Flow (cfs)
shorter duration
(1 day) is more
extreme
N
o
t
e
:
a
v
e
r
a
g
e
f
l
o
w
e
a
s
i
e
r
t
o
p
l
o
t
t
h
a
n
v
o
l
u
m
e
–
c
u
r
v
e
s
a
r
e
c
l
o
s
e
r
1DAY
3DAY
7DAY
15DAY
30DAY
47
48
Note: it’s important to define the year
define a year that
won’t break up longer
high flow averages
what about for low
flows?
48
49
Synthetic Events
Create synthetic, low-frequency event hydrographs to
model extremes, or other watershed conditions
(e.g. regulated flow)
1.
perform volume frequency analysis
•
produce the family of
m-day average flow frequency curves
2.
choose an historical event as a hydrograph “shape”
3.
scale the hydrograph to match flow of chosen exceedance
probabilities (e.g., the 1% chance event)
•
read the average flow for each duration for the chosen
exceedance probabilities
•
scale the hydrograph up or down to match that avg flow
49
50
Percent Chance Exceedance
Average Flow (cfs)
shorter duration
(1 day) is more
extreme
1/200-year
1DAY
3DAY
7DAY
15DAY
30DAY
Max Flow at Folsom Volume-Freq Curves
50
Rescaling Historical Event
51
1997 flood event
0
.
9
7
%
51
52
Rescaling Historical Event
1997 flood event, scaled to 3-day 1-in-200%
0
.
9
7
%
52
Modeled Operations for actual 1997
53
Storage
Level
115,000 cfs
Release
160,000 cfs
Inflow
53
Modeled Operations for 200-yr 1997
54
Release
Inflow
115,000 cfs
160,000 cfs
Storage
Level
54
Graphical Curve is Better for this
200
500
Cumulative Exceedance Probability
55
1986 flood event, scaled to 3-day 1-in-200%
Rescaling Historical Event
56
57
Graphical Curve is Better for this
57
2
0
0
5
0
0
C
u
m
u
l
a
t
i
v
e
E
x
c
e
e
d
a
n
c
e
P
r
o
b
a
b
i
l
i
t
y
Graphical Curve is Better for this
Cumulative Exceedance Probability
57
Volume Frequency Analysis
Sometimes there’s an additional step needed in the
volume frequency analysis…
3.
Check curves for consistency
•
Curves should not cross
•
Smooth by plotting log moments vs duration, or
versus each other
58
58
59
When volume-frequency curve require adjustment
N
o
t
e
:
a
v
e
r
a
g
e
f
l
o
w
e
a
s
i
e
r
t
o
p
l
o
t
t
h
a
n
v
o
l
u
m
e
–
c
u
r
v
e
s
c
l
o
s
e
r
59
60
To prevent curves
from crossing…
60
61
To prevent curves
from crossing…
61
62
Resulting Volume Duration Frequency Curves
N
o
t
e
:
a
v
e
r
a
g
e
f
l
o
w
e
a
s
i
e
r
t
o
p
l
o
t
t
h
a
n
v
o
l
u
m
e
–
c
u
r
v
e
s
c
l
o
s
e
r
62
peak
1-day
3-day
7-day
15-day
30-day
B
a
l
a
n
c
i
n
g
a
c
r
o
s
s
m
u
l
t
i
p
l
e
d
u
r
a
t
i
o
n
s
63
64
Frequency Curves along a Mainstem
•
If performing frequency analysis for multiple sites along a
mainstem, also creating a “family” of frequency curves
•
The curves should also be
consistent
, with downstream
curves not less than upstream curves, at all frequencies
–
curves should not cross
–
might not be consistent initially – different periods of record
–
some locations have large event in record, other don’t…
•
Perform similar smoothing of distribution parameters
64
Low Flow Frequency Analysis
•
Based on
annual minimum
flows
•
Estimate the probability that m-day average flow will
not exceed
a given value
mQn: m consecutive day average,
n year return period
example: 7Q10, 7-day 10-year low-flow
There is a 10% chance that the 7-day
average low-flow will be
less than
this value in any year
•
As the duration increases to months, independence
between events becomes less likely (7-day is common)
65
66
High vs Low Flows
•
High flows
–
Can usually fit an
analytical distribution
function to
unregulated
high flows
–
Generally not to
regulated
high flows - need
graphical
•
Low flows
–
can be more non-linear (effects of groundwater, etc),
–
might only allow
graphical
(empirical) frequency curve
66
67
Regulated Low Flow Volume-Freq Curves
7DAY 10YEAR
7DAY
3DAY
1DAY
Average Flow (cfs)
Percent Chance
Non
-exceedance
note that
low
-flow curves are
closer than high-flow...
510
15DAY
Folsom Minimum Pool Frequency
68
Non-exceedance Probability
Folsom Minimum Pool Elevation (ft)
boat ramp at 385’
boat ramp at 320’
68
69
Topics
•
Annual Extremes
–
annual maximums or minimums, Bulletin 17B/C
–
Instantaneous flows or
longer duration avg flow/volume
•
Partial Duration (peaks over threshold)
•
Daily flows or stages
–
Duration Curves
–
Summary Hydrographs
•
Annual Volumes
–
Drought, Dependent random variables
69
No independence - persistence
independence
Assumption
independence
No independence - persistence
Annual Maximum Series (AMS)
70
Willamette River at Salem, Oregon - Unregulated
70
Partial Duration Series (PDS)
71
Choose not just the most extreme each
year, but all
INDEPENDENT
events that
fall beyond a defined threshold
Willamette River at Salem, Oregon - Unregulated
71
Peaks over Threshold
Peaks Over Threshold
•
Block maximum approach “throws
out” data
•
Some blocks have small maxima
–
Smaller than non-maxima in other
blocks
•
What if we consider
independent
local maxima?
72
Threshold
Threshold
Block Maxima
Block Maxima
Local Maxima
Local Maxima
Peaks Over Threshold
•
Zero or more local maxima per block
–
Count of peaks needs to be considered
•
Need to ensure local maxima sufficiently independent
•
No longer dealing with order statistics
–
Cumulative probability ≠ 1 – AEP
73
•
We are interested in the distribution of
excesses
:
–
Given a set of values that exceed a threshold,
–
What is the distribution of the excess?
Peaks Over Threshold
74
Peaks Over Threshold
Partial Duration Series
Counts
•
Threshold choice affects count distribution
•
Usually assumed to have a
Poisson Distribution
1,250 cfs
1,250 cfs
2,500 cfs
2,500 cfs
Distribution of Values
•
Model magnitude of values
greater than threshold
•
Tend to cluster close to
threshold
Threshold
Threshold
Partial Duration Series
Partial Duration Series
Generalized Pareto Distribution
79
Distribution of Values
•
Cumulative probability,
given the values exceed
the threshold
•
NOT AN AEP
NOT AN AEP
Threshold = 1,250 cfs
Threshold = 1,250 cfs
GPD Fit to PDS
GPD Fit to PDS
81
Empirical
Partial Duration Series (PDS)
Calculating plotting positions:
If there are a total of
k
events in N years, then the plotting position
for the smallest event (rank = k) is based on
k
/N
•
If
k
>
N
, might be greater than 1.0
•
The plotting position for this event is interpreted, not as
exceedance probability, but as average of
k
/N
events per year
Then the plotting position estimate states
that on the average it is expected that 2.0
events per year will exceed 1000 cfs
Example
Smallest event = 1000
k=50, N=25
can’t interpret as probability…
81
82
Empirical
Frequency Curves: Willamette at Salem
partial duration vs annual max
p
a
r
t
i
a
l
d
u
r
a
t
i
o
n
s
e
r
i
e
s
a
n
n
u
a
l
m
a
x
i
m
u
m
s
e
r
i
e
s
annual peak flow (cfs)
can only plot
the top N
points on a
probability
axis
N=years
82
83
Topics
•
Annual Extremes
–
annual maximums or minimums, Bulletin 17B/C
–
Instantaneous flows or
longer duration avg flow/volume
•
Partial Duration (peaks over threshold)
•
Daily flows or stages
–
Duration Curves
–
Summary Hydrographs
•
Annual Volumes
–
Drought, Dependent random variables
83
No independence - persistence
independence
Assumption
independence
No independence - persistence
84
Flow/Stage Duration Curve
Fraction of days exceeding a given value, referred to
as
percent of time exceeded
OR
Estimate of the likelihood that a
daily flow
will exceed a
particular level
(refers to future, not soon…)
•
Graphical display of stream characteristics
•
Navigation, run-of-river hydropower, water supply
84
Flow- or Stage-Duration Curves
Plot of Flow vs Percent of time exceeded
(see Maidment, 1992, pg. 8.27, 18.53, 27.40)
Computation
1.
Rank all the daily flows for period of record from largest to
smallest
2.
Percent of time exceeded for each = rank
total number of
days in record
85
i.e.,
relative
frequency
86
Starting with a daily flow record…
Question:
How
much of the
time can we
expect future
flows to be
between 2,000
and 4,000 cfs?
Interested in hydropower generation
86
How often were
past flows
between 2,000 and
4,000 cfs?
87
2000
4000
87
Starting with a daily flow record…
Question:
How
much of the
time can we
expect future
flows to be
between 2,000
and 4,000 cfs?
How often were
past flows
between 2,000 and
4,000 cfs?
19%
70%
More than 4000
cfs is released
19% of the time,
and
at least 2000 cfs
is released 70% of
the time,
so, 51% between
2000 and 4000
cfs
2000
4000
Order the data from low to high, and compute relative
frequency of exceedance
considering
all
flows, not
just annual extremes
… but lost event DURATION
88
88
19%
70%
Comparing Unregulated to Regulated
Regulated flows
are between
2000 and 4000
cfs 51% of the
time,
unregulated flows
are in the range
14% of the time.
44%
30%
American River flows, unreg and reg
2000
4000
89
89
90
Flow Duration Analysis Cautions
•
Data points are
not independent
, so frequency does not
approximate probability in the
near term
–
Only approximates probability in future
–
An estimate of
daily probability
, not annual
•
Analytical distributions do not fit data
•
Need sufficient record length to estimate distribution
tails
90
91
Topics
•
Annual Extremes
–
annual maximums or minimums, Bulletin 17B/C
–
Instantaneous flows or
longer duration avg flow/volume
•
Partial Duration (peaks over threshold)
•
Daily flows or stages
–
Duration Curves
–
Summary Hydrographs
•
Annual Volumes
–
Drought, Dependent random variables
91
No independence - persistence
independence
Assumption
independence
No independence - persistence
Summary “Hydrographs”
92
Rio Madeira @ Porto Velho, Brazil
1967 - 2005
Sept 1 only
min = 2630
max = 11614
50%
92
Summary “Hydrographs”
93
Rio Madeira @ Porto Velho
1967 - 2005
Sept 1 only
min = 2630
max = 11614
50%
93
Summary “Hydrographs”
94
Unregulated
Regulated
American River blw Folsom
94
Summary “Hydrographs”
95
Unregulated
Regulated
American River blw Folsom
95
Summary “Hydrographs”
96
Unregulated
Regulated
Columbia River at the Dalles
96
Summary “Hydrographs”
97
Unregulated
Regulated
Columbia River at the Dalles
97
98
Topics
•
Annual Extremes
–
annual maximums or minimums, Bulletin 17B/C
–
instantaneous
flows or longer duration avg flow/volume
•
Partial Duration (peaks over threshold)
•
Daily flows or stages
–
Duration Curves
–
Summary Hydrographs
•
Annual Volumes
–
Drought
, Dependent random variables
No independence - persistence
independence
98
Assumption
independence
No independence - persistence
Drought
•
Concerned with
long-term low flows
•
Annual streamflow volume is generally not independent
year to year
–
Connection to groundwater causes longer-term persistence
–
Levels that are depressed in a previous period will probably not
recover in time to produce normal stream flows in the next period
•
Droughts can be multiple-year-events, and consideration of
several years at a time
reduces the effective sample size
99
99
Dependent Random Variables
Annual extreme events are
independent
random variables
•
Distant enough to show no dependence on previous value
•
Can do standard frequency analysis on them
Droughts require analysis of flow sequence as a
dependent
random variable
•
Dependent on the previous period’s value
•
Dependence between consecutive periods is
serial correlation
•
Daily flow records are aggregated to weeks, months or years
depending on analysis purpose
100
100
or, auto-correlation
Time Series Model
•
Time series modeling is used to capture dependence in
sequences
Y
t
=
+
(Y
t-1
–
) +
t
•
Useful time series models have been developed for monthly
and annual flows but
NOT
for daily flows
101
101
t
~ N(0,
)
Drought Statistics
Duration
Magnitude
Severity
severity =
magnitude
duration
102
102
Drought Statistics
Sampling Error
•
We experience very few drought periods over the typical
record length
•
Small sample for drought statistics meaning large sampling
error
103
103
Stochastic Streamflow Models
•
Statistical time series model used to
characterize
streamflows,
and the same model is used to randomly
generate
probable
sequences of streamflows
–
Can use
all
time periods, not just droughts, to infer streamflow
statistics
(mean, variability, dependence/correlation)
•
Models can be used to infer drought statistics and estimate
probability of severe droughts
by generating long, random,
synthetic records with more droughts
•
Most common is Auto-Regressive lag1, AR(1)
•
We’ll see its use in the Monte Carlo Simulation lecture
104
104
105
Topics
•
Annual Extremes
–
annual maximums or minimums, Bulletin 17B/C
–
instantaneous
flows or longer duration avg flow/volume
•
Partial Duration (peaks over threshold)
•
Daily flows or stages
–
Duration Curves
–
Summary Hydrographs
•
Annual Volumes
–
Drought, Dependent random variables
No independence - persistence
independence
105
Assumption
independence
No independence - persistence
106
Topics
•
Annual Extremes
–
annual maximums or minimums, Bulletin 17B/C
–
Instantaneous flows or
longer duration avg flow/volume
•
Partial Duration (peaks over threshold)
•
Daily flows or stages
–
Duration Curves
–
Summary Hydrographs
•
Annual Volumes
–
Drought, Dependent random variables
106
No independence - persistence
independence
Assumption
independence
No independence - persistence
References
•
Corps of Engineers, 1993. Hydrologic Frequency Analysis, EM–1110-2-
1415, Department of the Army, Washington D.C.
•
Dracup, J.A., Lee K.S., and E.G. Paulson, Jr., 1980, On the Definition of
Droughts, Water Resources Research, April, v16(2), p297-302.
•
Hydrologic Engineering Center, 1985. Stochastic Analysis of Drought
Phenomena, TD 25, p 140.
•
Maidment, Davis R. (editor), 1992. Handbook of Hydrology, Mcgraw-
Hill, Inc., New York
.
107
107
This lecture is a bit of a review and connect the dots of some material already covered, a preview of a topic that’s coming up, and then some new topics that fit within the theme.
Lecture 1.5 Time-series Analysis
1.5 Time Series Analysis
This content delves into the methods and assumptions involved in studying hydrologic time-series data for flood frequency analysis. It covers topics such as different types of assumptions, including independence and persistence, and highlights how streamflow data can be analyzed to find annual maximum values. The material also discusses the insights that can be gained from studying time-series data, such as extracting information, making assumptions, estimating values or probabilities, and the types of analysis required for each aspect.
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Lecture 1.5 Analyzing Hydrologic Time-Series a road map Flood Frequency Analysis Hydrologic Engineering Center Beth Faber, PhD, PE
Goals Review (or preview) some ways we might study a time-series, and what we can learn from it 1. What information we extract 2. What assumptions we make 3. What values or probabilities we can estimate 4. What type of analysis each requires 2
Assumption Assumption Topics independence independence Annual Extremes annual maximums or minimums, Bulletin 17B/C instantaneous flows or longer duration avg flow/volume independence independence Partial Duration (peaks over threshold) Daily flows or stages Duration Curves Summary Hydrographs Annual Volumes Drought, Dependent random variables No independence No independence - - persistence persistence No independence No independence - - persistence persistence 3 3
Assumption Assumption Topics independence independence Annual Extremes annual maximums or minimums, Bulletin 17B/C instantaneous flows or longer duration avg flow/volume independence independence Partial Duration (peaks over threshold) Daily flows or stages Duration Curves Summary Hydrographs Annual Volumes Drought, Dependent random variables No independence No independence - - persistence persistence No independence No independence - - persistence persistence 4 4
Streamflow data, daily flow record American River at Fair Oakes (unregulated) American River at Fair Oaks (unregulated) 300000 108 years of daily unregulated streamflow data 250000 average daily flow (cfs) 200000 150000 100000 50000 0 Nov-04 Oct-14 Oct-24 Oct-34 Oct-44 Oct-54 Oct-64 Oct-74 Oct-84 Oct-94 Oct-04 reservoir constructed 5
Streamflow data, find annual max American River at Fair Oakes (unregulated) American River at Fair Oaks (unregulated) 300000 consider each year's maximum flow 250000 often, annual peaks are often, annual peaks are instantaneous instantaneous values values average daily flow (cfs) 200000 150000 100000 50000 0 Nov-04 Oct-14 Oct-24 Oct-34 Oct-44 Oct-54 Oct-64 Oct-74 Oct-84 Oct-94 Oct-04 6
American River, ANNUAL MAX SERIES, AMS annual maximum values are independent annual maximum values are independent 300000 108 years of annual peak flow data 250000 Annual Peak Streamflow (cfs) 200000 150000 100000 50000 0 1926 1931 1935 1939 1944 1948 1974 1978 1983 1987 1991 1905 1909 1913 1918 1922 1952 1957 1961 1965 1970 1996 2000 2004 2009 7 7
Assumptions What we have: series of annual maximum flows Treat these flows as a random, representative sample from the population of interest -- -- probability distribution of annual probability distribution of annual peak flow peak flow Generally, we assume the sample is IID annual peak flows are random and independent peak flows are identically distributed homogeneous data set sample is adequately representative of the population (but estimate of the distribution improves with sample size) NOTE, since we collect data over time, we re assuming the data/watershed is stationary . 8
Histogram divide data range into bins, count number of observations in each bin (frequency), divide by total (relative frequency) 0.45 0.41 0.4 0.33 0.35 0.3 relative frequency 0.25 0.2 count/obs 0.13 0.15 0.1 0.04 0.04 0.010.030.020.00 0.000.01 0.05 0 37.5 62.5 87.5 112.5137.5162.5187.5212.5237.5262.5 0 25 50 75 100 125 150 175 200 225 250 275 Flow, in thousands of CFS this shape is similar to a PDF, probability density function this shape is similar to a PDF, probability density function 9 9
Histogram divide data range into bins, count number of observations in each bin (frequency), divide by total (relative frequency) 0.25 0.21 0.2 0.18 0.18 switch switch to to log log10 (flow) 0.15 relative frequency 0.13 0.11 10(flow) 0.1 count/obs 0.07 0.06 0.04 0.05 0.02 0.010.00 0.01 0 3.4 3.6 3.8 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6 4.8 5.0 5.2 5.4 5.6 4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 Log (Flow, in thousands of CFS) Log Flow (thousands of CFS) this shape is similar to a PDF, probability density function this shape is similar to a PDF, probability density function 10 10
Definition of PDF and CDF 0.2 A Probability Density Function (PDF), f(x), defines the probability of occurrence for a continuous random variable. area under curve = probability area under curve = probability PDF Probability per Unit Value 0.15 0.1 p 0.05 0 X of Variable X Value The Cumulative Distribution Function (CDF), F(x) is the probability the random variable is less than some value curve = probability curve = probability 1 0.8 Probability Less Than CDF 0.6 0.4 0.2 p 11 0 11 X of Variable X Value
Empirical Cumulative Distribution based on observation based on observation Empirical or Graphical estimate: based on order statistics, using plotting position (estimated non-exceedance probability) Cumulative probability of observation magnitude is based on its position within the sample, and the sample size Each observation has incremental prob = 1/N mi relative relative frequency frequency i.e., i.e., N mi = rank of ordered observation i (m=1 as smallest value, N as largest) N = total # of observations P[X xi] = estimate probability that outcome will be smaller than observation x estimate probability that outcome will be smaller than observation xi i by the fraction of the observed sample that by the fraction of the observed sample that is is smaller than x smaller than xi i 12 12
Plotting Positions Want to avoid a value equal to 1 (N/N)! Some common plotting positions: m Weibull N + 1 p = mean estimate of probability p =m 0.3 Median N + 0.4 median estimate of probability Hirsch-Stedinger based on threshold-exceedance where m = rank N = total # of values
Empirical Cumulative Distribution plotting position plotting position Non-Excdnc Probability Non-Excdnc Probability Year Rank Flow Year Rank Flow Plotting Position: 1997 108 252431 0.99 1955 14 10528 0.13 1955 107 189073 0.98 1975 13 10389 0.12 1964 106 183242 0.97 1972 12 10046 0.11 General Formula P =m + a 1986 105 170960 0.96 2008 11 9150 0.10 2005 104 161731 0.95 1966 10 8659 0.09 1963 103 152614 0.94 1939 9 8500 0.08 1950 102 132000 0.94 1907 8 8460 0.07 N + b 1980 101 124915 0.93 2001 7 8045 0.06 1928 100 119000 0.92 1931 6 7920 0.06 Weibull: a = 0.0, b = 1.0 Median: a = -0.3, b = 0.4 1982 99 113126 0.91 1990 5 7606 0.05 1907 98 105000 0.90 1961 4 6914 0.04 1909 97 98000 0.89 1988 3 5447 0.03 1970 96 88316 0.88 1994 2 5009 0.02 1969 95 83526 0.87 1977 1 2359 0.01 14
Empirical Cumulative Distribution Exceedance Probability Estimated from Plotting Positions 1.00 cumulative non-exceedance probability 0.90 0.80 0.70 CDF = cumulative CDF = cumulative distribution function distribution function 0.60 0.50 0.40 0.30 0.20 0.10 0.00 0 50000 100000 150000 200000 250000 300000 Annual Peak Flow (cfs) 15 15
Empirical Cumulative Distribution Exceedance Probability Estimated from Plotting Positions 300000 for a Corps for a Corps frequency curve, frequency curve, we switch axes... we switch axes... 250000 annual peak flow (cfs) 200000 150000 100000 50000 0 0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 cumulative non-exceedance probability 16 16 annual
Empirical Cumulative Distribution also called a frequency curve also called a frequency curve Exceedance Probability Estimated from Plotting Positions 300000 250000 ...and adjust axis scales ...and adjust axis scales annual peak flow (cfs) 200000 150000 100000 50000 Normal probability scale Normal probability scale 0 0 0.01 0.05 0.1 0.2 0.5 0.8 0.9 0.95 0.99 1 cumulative non-exceedance probability 17 17 annual
Empirical Cumulative Distribution also called a frequency curve also called a frequency curve Exceedance Probability Estimated from Plotting Positions 1000000 log log scale scale ...and adjust axis scales ...and adjust axis scales annual peak flow (cfs) 100000 10000 Normal probability scale Normal probability scale 1000 0 0.01 0.05 0.1 0.2 0.5 0.8 0.9 0.95 0.99 1 cumulative non-exceedance probability 18 18 annual
Empirical Cumulative Distribution also called a frequency curve also called a frequency curve Exceedance Probability Estimated from Plotting Positions 1.01 Return Period 5 100 2 20 10 1000000 log log scale scale largest in 108 years, probability 1/108 0.9% equaled or exceeded 11 times in 108 years 10% switch to switch to exceedance exceedance annual peak flow (cfs) 100000 Weibull would Weibull would be 1/109 be 1/109 10000 0.9% Normal probability scale Normal probability scale 1000 1 0.99 0.95 0.9 0.8 0.5 0.2 0.1 0.05 0.01 0 cumulative exceedance probability Cumulative Exceedance Probability 19 19 Annual
Empirical Graphical Cumulative Distribution Exceedance Probability Estimated from Plotting Positions 1.01 Return Period 5 100 2 20 10 1000000 log log scale scale annual peak flow (cfs) 100000 10000 Normal probability scale Normal probability scale 1000 1 0.99 0.95 0.9 0.8 0.5 0.2 0.1 0.05 0.01 0 cumulative exceedance probability Cumulative Exceedance Probability 20 20 Annual
for unregulated, for unregulated, instantaneous annual instantaneous annual maximums maximums Analytical Distribution Exceedance Probability Estimated from Plotting Positions 1.01 Return Period 5 100 2 20 10 1000000 log log scale scale Log Pearson III per Bulletin 17B/C annual peak flow (cfs) 100000 10000 Normal probability scale Normal probability scale 1000 1 0.99 0.95 0.9 0.8 0.5 0.2 0.1 0.05 0.01 0 cumulative exceedance probability Cumulative Exceedance Probability 21 21 Annual
Distribution Fitting a model for probability: Step 1: Choose Model Step 2: Calibrate Model (Estimate Distribution Parameters) Can estimate population parameters using sample statistics. The general descriptive parameters are: Distribution Moments Central Tendencywhere? Dispersion or Spreadhow wide? where? (location) (location) how wide? (scale) (scale) (shape) (shape) Asymmetry symmetrical? symmetrical? PDF 22
Sample Standard Deviation Sample Skew Coefficient Sample Mean N N N Xi X3 N X =1 1 Xi X2 N i=1 Xi S = N 1 g = i=1 (N 1)(N 2) S3 i=1 average distance average distance from mean from mean where: Xi = sample member, i N = sample size dimensionless dimensionless g g = 0 has same unit has same unit as variable as variable g g < 0 S g g > 0 23
Distribution Moments First moment: mean X = N i=1 1 NXi Second central moment: variance Sx Xi X2 1 N 2= N 1 i=1 Third standardized central moment: skew gx= N 1 N 2 Sx N 1 N Xi X3 3 i=1 24
Distribution Parameters CDF, frequency curve form CDF, frequency curve form 1400000 1200000 1000000 Streamflow (cfs) Peak difference in difference in the mean the mean 800000 600000 400000 difference in the difference in the standard deviation standard deviation 200000 0 99 95 90 75 50 25 10 5 1 0.1 Exceedence Frequency 100 x probability of exceeding 100 x probability of exceeding 25
Distribution Parameters CDF, frequency curve form CDF, frequency curve form 1400000 1200000 1000000 Streamflow (cfs) Peak negative negative skew = 0 is skew = 0 is straight line straight line on this on this Normal prob Normal prob axis axis 800000 changing changing the skew the skew 600000 400000 positive positive 200000 0 99 95 90 75 50 25 10 5 1 0.1 Exceedence Frequency 100 x probability of exceeding 100 x probability of exceeding 26
for unregulated, for unregulated, instantaneous annual instantaneous annual maximums maximums Analytical Distribution Exceedance Probability Estimated from Plotting Positions 1.01 Return Period 5 100 2 20 10 1000000 log log scale scale Log Pearson III per Bulletin 17B/C annual peak flow (cfs) 100000 10000 Normal probability scale Normal probability scale 1000 1 0.99 0.95 0.9 0.8 0.5 0.2 0.1 0.05 0.01 0 cumulative exceedance probability Cumulative Exceedance Probability 27 27 Annual
Bulletin 17B dated March 1982 Bulletin 17C dated Feb 2018 28 28
B17B/C: Frequency Analysis of Annual Peak Flows Estimate LogPearson III distribution for unregulated annual peak flows Method of Moments moments of sample used to estimate moments of distribution Data challenges: Missing Flows (Broken or Incomplete Record) Low and High Outliers Zero Flows Additional information to improve estimates: Historical/Paleo Information Regional Skews (use a weighted skew) Allowing a longer record site to improve short record estimate many of the methods many of the methods for handling data for handling data challenges and using challenges and using additional information additional information are improved in are improved in Bulletin 17C Bulletin 17C 29 29
for unregulated, for unregulated, instantaneous annual instantaneous annual maximums maximums Analytical Distribution Exceedance Probability Estimated from Plotting Positions 1.01 Return Period 5 100 2 20 10 1000000 log log scale scale Log Pearson III per Bulletin 17B/C annual peak flow (cfs) 100000 90% confidence interval 10000 Normal probability scale Normal probability scale 1000 1 0.99 0.95 0.9 0.8 0.5 0.2 0.1 0.05 0.01 0 cumulative exceedance probability Cumulative Exceedance Probability 30 30 Annual
Also Interested in Regulated Flow Reservoir Storage Level Inflow 160,000 cfs 115,000 cfs Release 31 31
Streamflow data, regulated American River below Folsom Reservoir 300000 period of record 1955 - 2009 homogeneous homogeneous = = identically identically- - distributed distributed = = stationary stationary 250000 Is this regulation homogeneous? Is this regulation homogeneous? average daily flow (cfs) 200000 150000 100000 50000 0 Nov-04 Oct-14 Oct-24 Oct-34 Oct-44 Oct-54 Oct-64 Oct-74 Oct-84 Oct-94 Oct-04 reservoir constructed 32 32
Streamflow data, regulated American River below Folsom Reservoir 300000 period of record 1921 - 2002 250000 2 benefits: 2 benefits: these are simulated outflows: these are simulated outflows: average daily flow (cfs) 200000 homogeneous homogeneous operation, operation, and longer and longer period period HOMOGENEOUS operation HOMOGENEOUS operation 150000 100000 50000 0 Nov-04 Oct-14 Oct-24 Oct-34 Oct-44 Oct-54 Oct-64 Oct-74 Oct-84 Oct-94 Oct-04 33 33
Streamflow data, regulated American River below Folsom Reservoir 300000 consider each year's maximum flow 250000 regulated annual peaks regulated annual peaks average daily flow (cfs) 200000 150000 100000 50000 0 Nov-04 Oct-14 Oct-24 Oct-34 Oct-44 Oct-54 Oct-64 Oct-74 Oct-84 Oct-94 Oct-04 34 34
Folsom Release, ANNUAL MAX annual maximum values are independent annual maximum values are independent 300000 300000 108 years of annual peak flow data 81 years of annual peak flow data 250000 250000 Annual Peak Streamflow (cfs) Annual Peak Streamflow (cfs) 200000 200000 150000 150000 100000 100000 50000 50000 0 0 1926 1931 1935 1939 1944 1948 1974 1978 1983 1987 1991 1905 1909 1913 1918 1922 1926 1931 1935 1939 1944 1948 1952 1957 1961 1965 1970 1974 1978 1983 1987 1991 1996 2000 2004 2009 1905 1909 1913 1918 1922 1952 1957 1961 1965 1970 1996 2000 2004 2009 35 35
Plotted Annual Maximums Exceedance Probability Estimated from Plotting Positions 1.01 Return Period 5 100 2 20 10 1000000 annual peak flow (cfs) 100000 10000 1000 1 0.99 0.95 0.9 0.8 0.5 0.2 0.1 0.05 0.01 0 cumulative exceedance probability Cumulative Exceedance Probability 36 36
Does an Analytical Curve Make Sense? Exceedance Probability Estimated from Plotting Positions 1.01 Return Period 5 100 2 20 10 1000000 annual peak flow (cfs) 100000 10000 1000 1 0.99 0.95 0.9 0.8 0.5 0.2 0.1 0.05 0.01 0 cumulative exceedance probability Cumulative Exceedance Probability 37 37
Graphical Curve is Better for this Exceedance Probability Estimated from Plotting Positions 1.01 Return Period 5 100 2 20 10 1000000 annual peak flow (cfs) 100000 10000 1000 1 0.99 0.95 0.9 0.8 0.5 0.2 0.1 0.05 0.01 0 cumulative exceedance probability Cumulative Exceedance Probability 38 38
Graphical Curve is Better for this Exceedance Probability Estimated from Plotting Positions 1.01 Return Period 5 100 2 20 10 1000000 annual peak flow (cfs) 100000 how do we how do we extrapolate? extrapolate? 10000 1000 0.99 0.95 0.9 0.8 0.5 0.2 0.1 0.05 0.01 cumulative exceedance probability Cumulative Exceedance Probability 39 39
Extending a Regulated Curve To extend a regulated frequency curve, we can construct synthetic events with a specified frequency, e.g. 100-yr, 200-yr, etc Estimate frequency curves for longer duration average flows or volumes (3-day, 7-day, 30-day, etc) Construct hydrograph for a given frequency Route through operation model to get peak outflow NOTE: will assume likelihood of peak release is related to likelihood of peak inflow 4040
Graphical Curve is Better for this Exceedance Probability Estimated from Plotting Positions 1.01 Return Period 5 100 2 20 10 200 500 1000000 annual peak flow (cfs) 100000 10000 1000 0.99 0.95 0.9 0.8 0.5 0.2 0.1 0.05 0.01 cumulative exceedance probability Cumulative Exceedance Probability 41 41
Assumption Assumption Topics independence independence Annual Extremes annual maximums or minimums, Bulletin 17B/C Instantaneous flows or longer duration avg flow/volume independence independence Partial Duration (peaks over threshold) Daily flows or stages Duration Curves Summary Hydrographs Annual Volumes Drought, Dependent random variables No independence No independence - - persistence persistence No independence No independence - - persistence persistence 42 42
Longer Durations of Flow/Volume Volume Frequency Analysis Create a family of frequency curves for a given site that specify average flow across various durations 1. Compute m-day moving average TS for various durations, m m might be 1-day, 5-day, 30-day etc. 2. Extract annual maximums for each duration 3. Perform frequency analysis for each duration Similar to Bulletin 17B/C procedure, but not instantaneous peak Compute plotting positions and distribution parameters from each period of record sample 43 43
Longer Durations Max Average Flows 1-day AMS 1-day Daily Average 5-day AMS 5-day Daily Average 10-day AMS 10-day Daily Average 44 44
Longer Duration Max Average Flows 120000 Kootenai River @ Libby daily 1day Unregulated Streamflow (cfs) 100000 15day 30day 80000 60day 90day 60000 40000 20000 0 1-Feb-70 2-Aug-70 31-Jan-71 1-Aug-71 30-Jan-72 30-Jul-72 28-Jan-73 29-Jul-73 27-Jan-74 28-Jul-74 4545
Max Flow at Folsom Volume-Freq Curves Note: average flow easier to plot than volume curves are closer 1DAY 3DAY 7DAY 15DAY 30DAY shorter duration shorter duration (1 day) is more (1 day) is more extreme extreme Percent Chance Exceedance 47 47
Note: its important to define the year define a year that won t break up longer high flow averages what about for low flows? 48 48
Synthetic Events rare rare Create synthetic, low-frequency event hydrographs to model extremes, or other watershed conditions (e.g. regulated flow) 1. perform volume frequency analysis produce the family of m-day average flow frequency curves 2. choose an historical event as a hydrograph shape 3. scale the hydrograph to match flow of chosen exceedance probabilities (e.g., the 1% chance event) read the average flow for each duration for the chosen exceedance probabilities scale the hydrograph up or down to match that avg flow 49 49
Max Flow at Folsom Volume-Freq Curves 1/200-year 1DAY 3DAY 7DAY 15DAY 30DAY shorter duration shorter duration (1 day) is more (1 day) is more extreme extreme Percent Chance Exceedance 50 50
Rescaling Historical Event 1997 flood event 500000 American River @ Folsom 450000 Unregulated Streamflow (cfs) 400000 350000 1997 hourly 1day 300000 0.6% chance exc 3day 250000 7day 15day 200000 1.0% 0.97% 150000 0.8% 100000 1.2% 50000 0 12/19 12/23 12/27 12/31 1/4 1/8 1/12 1/16 51 51
