Prediction of Watershed Runoff

1
Prediction of Watershed Runoff

• Tom Dunne
• Winter 2008

2
Intent of Computer Modeling in the Course (Type A)

• Only very few of you are likely to become
  modelers or users of software developed by
  agencies, consulting firms, or academics.
• Most of this activity involves software developed
  or promoted by agencies to discharge their
  regulatory function and control/standardize/facili
  tate outside design work that they must evaluate.
  
• Most public-access agency software now being made
  user-friendly and marketed, including by private
  firms
• You will eventually become familiar with these
  and other computer models to a degree which is
  not possible in a course.
• The labs in this course will introduce you to
  examples of what is available and how careful,
  thorough, and insightful application is
  necessary. These things will run themselves, but
  not necessarily intelligently.

3
Intent of Computer Modeling in the Course (Type B)

• Most of you will not become modelers
• Not a management function
• Not high-status (just inaccessible!). Extent to
  which some modelers pull wool over managers eyes
  remains impressive.
• The servant of management
• Allows examination of the implications of various
  policy or design options
• Managers need some appreciation of what modeling
  involves, and what modelers think they are doing
  (both your own agencys and a consulting firms)
• Managers need to understand the limits of
  modeling applicability --- what is the basic
  conceptual model underlying the construction of
  the model? Is it applicable to the situation
  under review? What is the expected reliability
  of predictions? What can be done to assure
  intelligent application, improved reliability,
  diligent application over the realistic range of
  conditions, and transparency?
• Managers and policy-makers should not frame a
  policy or management question the resolution of
  which requires predictions of an unattainable
  level of precision.

4

• For these reasons, we are going to introduce you
  to various forms of hydrologic modeling for doing
  the most widely applied tasks in water resources
  management
• Remember that there is a large number of models
  for doing each task, and new ones are being
  generated continually --- though new conceptual
  models are very rare.
• So, dont grab on to any model as the way of
  making calculations. If two models give
  radically different answers, work to reconcile or
  resolve the differences. There is a reason for it
  --- may lie in choice of parameter values,
  inappropriateness of one model for the
  conditions, etc.
• Differences in predictions often result from
  decisions made about how to set up and apply the
  models, and the different questions being asked
  or emphasized by competing interests.

5
Prediction of watershed storm runoff
What do we want to know?

• Volumes of storm runoff
•  Entire storm hydrographs
• Continuous simulation of streamflow (storm and
  dry-weather flow)
• Deterministic prediction of peak rates of runoff
  from small watersheds
• Probabilistic prediction of peak flows (from any
  size of watershed)

6
---- There is a quasi-infinite number of methods
of predicting watershed storm runoff----
Increasingly codified, computerized, and promoted
with acronyms such as SWMM, SWAT, other
four-letter words are available---- But all
based on one of a a few concepts

• Soil moisture accounting (SMA) (Like water
  balance from ESM 203 applied to short time
  periods)
• Constant speed of runoff over a plane
• A Curve Number index of watershed
  responsiveness to rainfall
• Some are lumped models (basin is a single
  space)
• Some are distributed models (represent spatial
  variation of watershed characteristics and runoff
  itself)

7
Prediction of storm runoff volume (expressed as
depth of runoff --- i.e. volume per unit
area)Thompson, 1999, Hydrology for Water
Management

• Baseflow is added to predicted storm flow, using
  the water balance method (see previous, or ESM
  203)

8
Prediction of Volume of Flood Runoff

• Need some mechanism ( a runoff model) to convert
  a portion of the rain/melt into runoff to the
  channel system, and of representing the empirical
  observation that this proportion tends to
  increase through time during a storm or season as
  the watershed becomes wetter (stores more water)
• i.e. there is a feedback between the water stored
  in a watershed and the proportion of rain that is
  converted to runoff

9
Runoff models to choose

• Calculate the entire water balance including
  quickflow
• Precipitation-Runoff Modeling System (US
  Geological Survey)
• HEC-Hydrologic Modeling System (US Army Corps of
  Engineers)
• Soil Water Assessment Tool (US Dept of
  Agriculture)
• BASINS 3.0 (US Environmental Protection Agency)
• In lab we will concentrate on a use of HEC-HMS
  that emphasizes the quickflow component, and so
  is useful mainly for tributaries that have little
  or no baseflow
• The full models are combinations of the
  water-balance approach practiced in ESM 203 and
  several options for computing quickflow.

10
Representation of runoff in HEC-HMS, promoted by
the US Army Corps of Engineers
11
HEC-HMS uses separate sub-models to compute each
component of runoff

• Runoff volume (per storm or per day/month)
• Timing of direct runoff (quickflow)
• Baseflow (delayed flow)
• Speed and timing of channel and floodplain flow
  to a river basin outlet

12
There are options (models) for each step in
runoffe.g. for computing runoff volume

• Initial abstraction and constant loss
  (infiltration, as expressed by the Findex in
  previous lecture)
• Green-Ampt infiltration model
• Soil Conservation Curve Number
• Individual rainstorms
• small basins or Hydrologic Response Units (HRUs)
• Soil Moisture Accounting (SMA)
• Water balance model
• Best for continuous modeling through many time
  steps in large basins or their HRUs or pixels
• Gridded SMA

13
There are options (models) for each step in
runoffe.g. for computing runoff volume

• Initial abstraction and constant loss
  (infiltration, as expressed by the Findex in
  previous lecture)
• Green-Ampt infiltration model
• Soil Conservation Curve Number
• Individual rainstorms
• small basins or Hydrologic Response Units (HRUs)
• Soil Moisture Accounting (SMA)
• Water balance model
• Best for continuous modeling through many time
  steps in large basins or their HRUs or pixels
• Gridded SMA

http//www.ce.utexas.edu/prof/maidment
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There are options (models) for each step in
runoff e.g. for computing runoff volume

• Initial abstraction and constant loss
  (infiltration)
• Green-Ampt infiltration model
• Soil Conservation Service Curve Number
• Individual rainstorms
• small basins or HRUs
• Soil Moisture Accounting (SMA)
• Water balance model
• Best for continuous modeling through many time
  steps in large basins or their HRUs or pixels
• Gridded SMA

Forested watershed in Danville, NE Vermont
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SCS method for prediction of storm runoff
(QKFLO) volume (R, as a depth per unit area)

• Very widely used in prediction software
• Accounts for effects of soil, properties, land
  cover, and antecedent moisture
• Prediction of storm flow depends on total
  rainfall rather than intensity
• Based on a very simple conceptual model, as
  follows.

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Prediction of storm runoff volume (SCS
method)All quantities expressed in inches of
water

• Total precipitation, P, is partitioned into
•  An initial abstraction, Ia , the amount of
  storage that must be satisfied before any flow
  can begin. This is poorly defined in terms of
  process, but is roughly equivalent to
  interception and the infiltration that occurs
  before runoff.
• --Thus, P Ia is the excess precipitation
  (after the initial abstraction) or the potential
  runoff.
•  Retention, F, the amount of rain falling after
  the initial abstraction is satisfied which does
  not contribute to the storm flow.
• Storm runoff Rs
• Thus P Ia F Rs

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• It is assumed that a watershed has a maximum
  retention capacity, Smax
• (1)
•  
• where F8 is the total amount of water retained
  as t becomes very large (i.e. in a long, large
  storm.) It is the cumulative amount of
  infiltration
•  
• It is also assumed that during the storm (and
  particularly at the end of the storm)
• (2)

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• The idea is that the more of the potential
  storage that has been exhausted (cumulative
  infiltration, F, converges on Smax), the more of
  the excess rainfall, or potential runoff,
  P-Ia, will be converted to storm runoff.
• The scaling is assumed to be linear.
• One more relationship that is known by
  definition
• (3)
• Combination of (2) and (3) leads to
• (4)

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• Another generalized approximation made on the
  basis of measuring storm runoff in small,
  agricultural watersheds under normal conditions
  of antecedent wetness is that
• (5)
• The few values actually tabulated in the
  original report are 0.15-0.2 Smax.
• Thus
• (6)

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• Combination of these relations yields
• (7)

for all PgtIa . ELSE R 0.

• Thus, the problem of predicting storm runoff
  depth is reduced to estimating a single value,
  the maximum retention capacity of the watershed,
  Smax.
• How to estimate Smax?

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Soil Conservation Service Storm Runoff
Relationship
Rs
P
22

• The entire rainfall-runoff response for various
  soil-plant cover complexes is represented by a
  single index called (with exquisite creativity!)
  the Curve Number.
• A higher curve indicates a large runoff response
  from a watershed with a fairly uniform soil with
  a low infiltration capacity.
• A lower curve is the smaller response expected
  from a watershed with a permeable soil, with a
  relatively high spatial variability in
  infiltration capacity.
• SCS developed an index of storm-runoff
  generation capacity, (the Curve Number), which
  would vary from 0 to 100 (implying roughly the
  percent of effective rainfall or potential
  runoff that is converted to flood runoff).

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• This curve number was then related to
  back-calculated values of Smax (inches) from
  measured storm hydrographs and equation (2) above
  to yield a relationship of the form
• or
• or

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CNs were then evaluated for many watersheds and
related to

• soil type (SCS soil types classified into Soil
  Hydrologic Groups on the basis of their measured
  or estimated infiltration behavior)
• vegetation cover and or land use practice 
• antecedent soil-moisture content
• A spatially weighted average CN is computed for a
  watershed.

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Hydrologic Soil Groups are defined in SCS County
Soil Survey reports
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Classification of hydrologic properties of
vegetation covers for estimating curve
numbers(US Soil Conservation Service, 1972)
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Runoff Curve Numbers for hydrologic soil-cover
complexes under average antecedent moisture
conditions
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Curve Numbers for urban/suburban land covers (US
Soil Conservation Service , 1975)
Hydrologic Soil Groups are defined in SCS County
Soil Survey reports
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SCS Curve Number Method

• No consideration is given to rainstorm intensity
  or duration.
• Method can be applied successively to parts of a
  rainstorm, and volume of runoff could be
  calculated separately for each increment.
  HEC-HMS does this in your lab exercise
• the form of the rainfall mass curve that is
  imagined by the user makes a very large
  difference to the predicted volume of runoff and
  its peak rate.
• orographic influences on rainfall also have a
  critical effect on predicted runoff volumes. See
  lab exercise

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SCS Curve Number Method

• No guidance given about the watershed size to
  which the method is applicable, except that the
  empirical relations were established for small
  watersheds.
• Now that the method is computerized, it is
  relatively easy to separate a watershed into
  sub-watersheds, and to the runoff calculation for
  each one separately, and then combine the runoff
  hydrographs that result (see lab exercise)

31

• Since rainfall is the largest term in any
  hydrologic calculation, its estimation is
  critical to runoff predictions
• No consideration is given to rainstorm intensity
  or duration.
• Method can be applied successively to parts of a
  rainstorm, and volume of runoff could be
  calculated separately for each increment.
  HEC-HMS does this in your lab exercise
• The form of the rainfall mass curve that is
  imagined by the user makes a very large
  difference to the predicted volume of runoff and
  its peak rate (about which more later).
• Orographic influences on rainfall also have a
  critical effect on predicted runoff volumes. See
  lab exercise

32

• No guidance given about the watershed size to
  which the method is applicable, except that the
  empirical relations were established for small
  watersheds.
• Now that the method is computerized, it is
  relatively easy to separate a watershed into
  sub-watersheds, and to the runoff calculation for
  each one separately, and then combine the runoff
  hydrographs that result (see lab exercise)

33
 Where tested against measured storm runoff
volumes, method is notoriously inaccurate. BUT

• 1. Method entrenched in runoff prediction
  practice and is acceptable to regulatory agencies
  and professional bodies. 2.   Attractively
  simple to use. 3.   Required data available in
  SCS county soil maps in paper and digital
  form. 4.   Method packaged in handbooks and
  computer programs 5.   Appears to give
  reasonable results --- big storms yield a lot
  of runoff, fine-grained, wet soils, with thin
  vegetation covers yield more storm runoff in
  small watersheds than do sandy soils under
  forests, etc. 6.   No easily available
  competitor that does any better. The method is
  already hidden in various larger computer
  models, such as HEC-HMS). 7. The task for a
  watershed analyst or regulator is to decide how
  to interpret and use the results.

34
There are options (models) for each step in
runoff e.g. for computing runoff volume

• Initial abstraction and constant loss
  (infiltration)
• Green-Ampt infiltration model
• Soil Conservation Curve Number
• Individual rainstorms
• small basins or HRUs
• Soil Moisture Accounting (SMA)
• Water balance model
• Best for continuous modeling through many time
  steps in large basins or their HRUs or pixels
• Gridded SMA

35
Soil Moisture Accounting Method (SMA) in HEC-HMS
36
Prediction of Flood Runoff

• Problem is that many runoff processes act
  simultaneously in a large basin and we have no
  hope of specifying the conditions affecting all
  of them at all times in a large diverse basin.
• Instead we use a simplified statement of runoff
  from each HRU or cell in a tributary watershed,
  such as a short-time-step (1-30 day) water
  balance

37
Prediction of runoff volume (R) generated during
a time stepin a Hydrologic Response Unit
38
Timing of stormflow runoff

• The various procedures outlined above calculate
  the volume of storm runoff, which in general must
  be added to the base flow, calculated by the
  soil-water balance or recession-curve method ESM
  203.
• We call this amount what to route.
• We call the topic of calculating the time
  distribution of runoff how to route.
• These are the two components of runoff hydrograph
  prediction

39
Options for routing flow down a channel network
in HEC-HMS

• Lag --- constant flow velocity
• Puls storage reservoir method
• views each channel reach as a small reservoir
• Muskingum method
• Kinematic wave
• Many others

40
Verify predictions wherever possible by
comparison with measured hydrographs
41
CalibrationAdjust values (parameters) in
various components of the models until prediction
fits observation well enough
42
How to route three options for a conceptual
model (1)

• Lag the water by a fixed time after it is
  generated. OK for small watersheds

Rs in each time unit
600 meters of channel/flow speed of 1 m/sec
Flow arrives 10 min after generation
43
How to route three options for a conceptual
model (2)

• View the watershed as an open book consisting of
  two planes and calculate the flow down these
  representative hillslopes, using Mannings
  equation and an equivalent roughness.
• Mannings roughness represents all the
  complications of the surface (including its
  spatial variability and flow paths) that will
  delay flow

v flow speed h flow depth s slope n
roughness
44
How to route three options for a conceptual
model (3)The unit hydrograph

• The fixed geometry of a watershed --- topography
  (gradients, elevation, effect on rainfall
  distribution), distribution of soil properties,
  channel network structure --- is the dominant
  control on the timing of storm runoff.
• A unit depth of storm runoff generated in a fixed
  time interval will always drain from the
  watershed at the same rate.
• This unit hydrograph can be estimated by
  superimposing and averaging storm runoff
  hydrographs, each of which has been reduced to a
  unit depth If total is 3 inches, divide all
  ordinates by 3
• The unit hydrograph is a characteristic of the
  watershed

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Unit hydrograph averaged from four recorded
hydrographs, normalized to one inch of
runoff(27.4 sq mi. watershed, Coshocton Ohio)
47
The Synthetic Unit Hydrograph of a watershed

• We could (it used to be done)
• derive a unit hydrograph for each of a sample of
  watersheds in a region
• correlate the various features of these
  hydrographs (e.g. peak discharge, lag to peak,
  duration) with watershed geometrical
  characteristics (e.g. area, steepness,)
• Use the resulting regression equation to estimate
  parameters of a synthetic unit hydrograph for
  ungauged basins
• A few regional regressions of this type were
  derived, before

48
The Synthetic Unit Hydrograph of a watershed

• Taking an even more abstract view of storm runoff
  timing, hydrologists concluded that the
  geometrical characteristics of most watersheds
  were sufficiently similar that a unit of storm
  runoff would drain from any landscape with
  approximately the same timing.
• Note, in their defense, these hydrologists were
  working in the rural US, and mostly in small
  watersheds
• This concept was enshrined in various
  approximations such as all unit hydrographs
  approximate triangles

49
The Soil Conservation Service Triangular
Synthetic Unit Hydrograph
P
Q
50
The Soil Conservation Service Triangular
Synthetic Unit Hydrograph
Q
51
The Soil Conservation Service Triangular
Synthetic Unit Hydrograph
52
Tests of the SCS unit hydrograph method

• 1600 runoff plots in SW US prediction of peak
  runoff greater than /- 50 in 67 of cases.
• 139 watersheds in E. Australia Marked lack of
  agreement between CN values obtained by
  conventional means and those back-calculated from
  recorded flows of previously chosen frequencies

53
Emerging Forms of Flood Prediction and Forecasting

• Higher resolution spatially distributed modeling
• Greater use of topographic information in
  hydrologic predictions because of availability of
  Digital Elevation Models
• Greater use of computerized spatial databases of
  watershed characteristics (soil, land use,
  channel networks, etc.)
• Greater use of satellite records of rainfall,
  radiation, temperature, etc. for driving energy-
  and water-balance calculations.

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Peak flows

• Can be predicted deterministically or estimated
  probabilistically (i.e. the risk of them can be
  imagined) 

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The Rational Runoff Formula

• Unspoken conceptual model is Horton overland flow

t75
Watershed boundary
t45
t60
Isochrones of runoff
t15
t30

tequilib 75 minutes
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Rational runoff model of a hydrograph
Qpeak
Q
tequilib
t 0
t
57
How to estimate tequilib (also called the time
of concentration)?

• Various handbook empiricisms from the
  1940s-50s, like the formula used by the National
  Resources Conservation Service (former Soil
  Conservation Service)

Where L is channel length and H is basin relief
Where S is the average channel gradient
Origin of such formulas is difficult to discern
(second one based on 6 small agricultural
basins!), but they are accepted by the
engineering community as proven useful. Other
formulas separate the time of concentration of
the average hillslope length and add it to the
tequilib of the maximum channel length, both
obtained from Mannings equation. Another
approach is to make your own estimate based on
observations and calculations using Mannings
equation
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Derivation of runoff rate
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Computation of rational runoff hydrograph (1)

• For rainstorms with duration, t gt tequilib 
• Qpeak C I A  
• Read Water in Environmental Planning pp 298-305
  for an attempt to elucidate this equation.
• A area
• I rainfall intensity of the storm with duration
  tequilib.
  Its duration, and
  therefore the estimation of tequilib strongly
  affect the chosen value of I because of the
  strong inverse relationship between rainfall
  intensity and duration and frequency.
• C f(land surface condition). Represents the
  loss rate of rainfall to infiltration.

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Computation of rational runoff hydrograph

• For rainstorms with duration, t gt tequilib 
• Qpeak C I A  
• As if by magic . If I is in inches/hr, A in
  acres (!), Q will be in cu. ft./s for a
  dimensionless C. This confirms our confidence
  that God gave this equation to our forefathers
  along with feet and inches.

61
Metric Rational Runoff Formula is

• For rainstorms with duration, t gt tequilib 
• Qpeak 0.278C I A
•  
• Where Qpeak is in cu. m/sec
• I is in mm/hr
• A is in sq km.

62
Rational equation predicts maximum discharge
values for a given drainage area --- for
conditions when the whole area is contributing
runoff.Therefore, rational formula only used
for small watersheds
Suppose rainstorm only lasts for tend minutes
63
Choice of rainstorm intensity is critical, but
not arbitrary

• Suppose the watershed is rural, has an area of
  400 acres, and has a tequilib of 30 min.
• And suppose we are interested in calculating the
  peak discharge in the 100-yr rainstorm (see
  later)
• Choose the intensity for a 30-minute storm with a
  recurrence interval of 100 yr.
• But then suppose you are asked what the 100 year
  peak discharge will be if the watershed is
  urbanized and its tequilib is reduced to 20
  minutes.
• Choose the 20-min, 100 yr rainstorm intensity,
  which will be higher than c.
• A conservative approach to choosing the critical
  intensity for design is to calculate the peak
  discharge for a range of durations and choose the
  largest predicted value. (transparency)

64
Rational runoff coefficient, C, for land
surfaces(Amer. Soc. Civil Engrs.)
65

• In fact, C is not truly a constant, but varies
  with recurrence interval of storm.
• This is probably because infiltration capacity
  measured at a point varies spatially, and more
  intense, rarer storms bring a larger fraction of
  the watershed up to saturation.
• Most values are estimated for the 2- and 10-yr
  storms. For comparison, C10/C2 1.33 C100/C10
  1.50  
• Variation of C with recurrence interval can be
  estimated by plotting measured values of rainfall
  intensity (over the duration of tequilib) and of
  flood peak against recurrence interval.Otherwise
  use values of C tabulated in handbooks and
  textbooks.

66
Prediction errors for the Rational Runoff Formula
are very large

• Australian study 271 small basins
•  
• 63 gave errors of gt 50 42 gt100.
•  
• Locally calibrated version behaved much better,
  but requires a lot of data and work (i.e. no
  one wants to analyze data any more!).
• C values did not vary with watershed
  characteristics as much as the tables of data in
  handbooks would have one believe.
• Considerable judgment and experience are
  required in selecting satisfactory values of C
  for design
• Check values against observed flows

67
Probabilistic prediction of peak discharges

• What has happened and the frequency of events in
  a record are the best indicators of what can
  happen and its probability of happening in the
  future.
• Requires a streamflow record of peaks at a
  station.
• The record is analyzed to estimate the
  probability of flood peaks of various sizes, as
  if they were independent of one another I.e. no
  persistent runs of wet and dry years)
• And then is extrapolated to larger, rarer floods.
• BUT most hydrologic records are short and
  non-stationary (i.e. conditions of climate and
  watershed condition change during the recording
  period). Magnitude of this problem varies, but
  needs to be checked in each case.

68
Flood-frequency Analysis and Prediction

• Analysis of empirical records of a flood at a
  place on a river network
• i.e. point-based, rather than spatial
• Concerned with events at a place, rather than
  processes distributed over a watershed
• Cannot be used for estimating the effects of
  environmental change on floods.
• Concept of stationarity is crucial. Mmm!

69
Basic IdeaApplication of your PStat course

• Because of the hydroclimatology and watershed
  conditions upstream of a point, the probability
  distribution of floods to be expected at the
  point can be estimated from the frequency
  distribution of past floods recorded there
• The observed frequency distribution is a sample
  from which the parameters of a theoretical
  probability distribution fitting the observations
  can be estimated
• Once fitted to the observed record, the
  theoretical probability distribution can be used
  to estimate the probabilities of other
  hypothetical expected discharges, either through
  interpolation or extrapolation of the recorded
  range of flows.

70
Probabilistic prediction of peak discharges

• Data used are the annual-maximum flow series
  the list of the largest flow of each year in a
  record of length n years.
• Annual-maximum instantaneous peak discharges and
  stages are available from the National Water Data
  Storage and Retrieval system (WATSTORE) at
  www.usgs.gov
• Arrange the flow values in descending order with
  rank m (largest rank 1).
• mI is the rank of the ith flood peak in a set of
  n peaks

the Weibull formula
71
Probabilistic prediction of peak discharges

• T is the recurrence interval (yr). The
  long-term average interval between floods greater
  than Qi
• Plot calculated values of T against Qi to
  develop a flood-frequency curve.
• See examples in Water in Environmental Planning,
  pp. 307-308.

72
Modifications

• Other plotting formulae are sometimes used
  instead of the Weibull formula
• They are chosen (and argued about) by their
  proponents to avoid a variety of numerical biases
  that arise when the data series are extrapolated
  to estimate rare flows.
• An example is the Cunnane formula

73
Probabilistic prediction based on 49 peak
discharges
?

• In principle, we could plot the two datasets on
  any kind of graph
• But if we intend to extrapolate to the relation
  to larger recurrence intervals, we need some more
  guidance
• We use theoretical probability distributions for
  this

74
Fitting curves to flood-frequency data for
extrapolation

• Several theoretical probability distributions fit
  the various observed frequency distributions of
  floods
• Each probability distribution can be represented
  by a straight-line fit to its cumulative form
  pQgtQi plotted on the appropriate graphical
  scale (analogous to the normal distribution graph
  paper in your PStat course).

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Probabilistic prediction of peak discharges

• Note that this procedure involves fitting a
  theoretical probability distribution to an
  observed sample drawn from an imaginary (but not
  well-understood) population
• The true theoretical probability distribution
  of flood discharges is not known, and we have no
  reason to believe it is simple or has only 1 or
  parameters.
• Plotting the data set on various types of graph
  paper with different scales, designed to
  represent various theoretical probability
  distributions as straight lines, yields graphs of
  different shapes, which when extrapolated beyond
  the limits of measurement predict a range of peak
  flood discharges.

77
Typical flood frequency curve(27.4 sq mi.
watershed at Coshocton Ohio)
78
Flood frequency plots of same record on different
probability papers
Log-extreme value paper Gumbel Type III
Log-probability paper
79
Probabilistic prediction of peak discharges

• Note that this procedure involves fitting a
  theoretical probability distribution to an
  observed sample drawn from an imaginary (but not
  well-understood) population
• There is no unique theoretical probability
  distribution of flood discharges, and we have no
  reason to believe it would be simple or have only
  1 or 2 parameters.
• Plotting the data set on various types of graph
  paper with different scales, designed to
  represent various theoretical probability
  distributions as straight lines, yields graphs of
  different shapes, which when extrapolated beyond
  the limits of measurement predict a range of peak
  flood discharges.

80
Bedient, P.B. and W.C. Huber, Hydrology and
Floodplain Analysis, (1992), Addison-Wesley Pub.
Co.
81
Plethora of proposed theoretical probability
distributions. How to choose?

• One common approach to choice of flood frequency
  plotting paper is to choose one on which the
  observed data plot as a straight line that can be
  extrapolated to estimate rare, large flood
  discharges. But .
• A second is to choose one of the common ones, but
  this still leads to different predictions among
  analysts
• So, in 1967, with later refinements, US
  Interagency Advisory Committee on Water Data
  published A Uniform Technique for Determining
  Flood Flow Frequencies, Bulletin 17B, US
  Geological Survey.

82
Faced with the dilemma that several probability
distributions might be chosen by different
analysts and used for extrapolation of the size
of rare floods

• So, in 1967 (updated in 1982), US federal
  agencies got together and decided that the
  theoretical probability distribution that most
  reliably fits observed annual-maximum flood
  frequencies is the Log Pearson Type III
  distribution. US Interagency Advisory Committee
  on Water Data published A Uniform Technique for
  Determining Flood Flow Frequencies, Bulletin
  17B, US Geological Survey. In 1982,
• Easiest way to fit such a flood-frequency curve
  to observed data is to obtain a sheet of the
  appropriate cumulative probability graph paper
  and plot each observed Qi against its calculated
  Ti and then draw a line (or a curve) through the
  data points.

83
Cookbook procedure for curve fitting of
log-Pearson Type III distribution to a flood
series from one station 1 (early steps should
be recognizable from your PStat class)

• Obtain all annual-maximum flows for a station
  from (for example www.usgs.gov)
• Download to Excel and rank them Q1,Qn.
• Convert each Qi to its logQi
• Use Excel to calculate Meanlog Qi , STDEV log
  Qi , SKEW log Qi , and Ti
• Convert Ti back to probability of exceedence, pi

84
Cookbook procedure for curve fitting of
log-Pearson Type III distribution to a flood
series from one station 2 (early steps should
be recognizable from your PStat class)

• Calculate the average of SKEW log Qi for the
  station, ---- called Cs
• A single-station value of Cs can be inaccurate
  and biased, so it is corrected using data from
  other stations in its region, according to
• Cw WCs (1-W)Cm
• Cw is the weighted skew coefficient
• W is a weighting factor
• Cs is the coefficient of skewness computed using
  the sample data,
• Cm is a generalized regional skewness, which is
  determined from a published map of the US

85
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86
Cookbook procedure for curve fitting of
log-Pearson Type III distribution to a flood
series from one station 3

• VarCm obtained from mapped values for US in
  Tech Bull 17B
• VarCs is obtained from
• A -0.330.88Cs if Cs 0.90 or
• A -0.520.30 Cs if Cs gt0.90
• B 0.94 0.26 Cs if Cs 1.50 or
• B 0.55 Cs gt1.50

87
Cookbook procedure for curve fitting of
log-Pearson Type III distribution to a flood
series from one station 4

• Then calculate the log Q for any Q value (such
  as your original Qi) from
• log Q MEANlog Qi KSTDEVlogQi
• K is a frequency factor that depends on Cs and Ti
  (tabulated in Bull. 17B)
• Plot the values of Q against T
• Extrapolate by choosing larger values of T and
  calculating Q

88
Sample frequency factor table (Haan, 1977)
89
Published example
Recurrence Interval (years)
90
Published example
Length of observed record used for illustration !!
91
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92
Bulletin 17Bhttp//acwi.gov/hydrology/Frequency/B
17bFAQ.html

• Instructions for plotting formula
• Fits data with a Log Pearson Type III
  distribution
• Instructions for how to estimate the skew of the
  distribution that should fit your station,
  based on regional skew patterns
• How to deal with outliers

93
Bulletin 17B Instructions for incorporating
historical information

• Overcome the short length of most flood
• Written records
• Flood marks chiseled on structures
• Dated tree scars (from tree rings)
• Dates sediment deposits
• Indicate maximum flood in n years, or number of
  floods greater than some stage or discharge in a
  fixed interval

94
Outliers

• Where should the outlier be plotted?
• Does it really represent the discharge with a
  70-yr recc. interval, or was it the 300-yr
  flood that fortuitously occurred in the 49-yr
  long record?

?
95
Bulletin 17B Instructions for incorporating
historical information

• Overcome the short length of most flood
• Written records
• Flood marks chiseled on structures
• Dated tree scars (from tree rings)
• Dates sediment deposits
• Indicate maximum flood in n years, or number of
  floods greater than some stage or discharge in a
  fixed interval

96
Uncertainty in assessing flood risk

• Short records (Bulletin 17B suggests using at
  least 10 years of record!)
• There is no fundamentally representative
  theoretical probability distribution. Policy for
  using (say) Log Pearson Type III is based on the
  assessment that applying it to many flood records
  yields minimum standard errors of estimate. But
  reasons that are not understood physically.
• Persistence problem
• Climate change
• Watershed change --- e.g how to assess the
  influence of the non-steady expansion of logging
  through the Oregon Cascades?
• Some changes are reversible (e.g. canopy
  re-establishment)
• Others are not (e.g. many roads and ditches)

97
Uncertainty in assessing flood risk

• So, use the accepted methodology (remember that
  the acceptability of these and similar techniques
  is based on professional agreements), and THEN
  for important decisions focus on the evidence for
  extreme events, even if you cant quantify their
  probability.
• Examine potential for non-hydrologic floods,
  or conditions that would aggravate a hydrologic
  flood
• Landslide dam-break flood
• Trestle bridge that could block floating woody
  debris

98
Recurrence interval (return period) of the T-year
flood

• Recurrence interval is the average interval
  between floods that are greater than a specified
  discharge
• E.g. if the probability of exceeding 20 m3/sec in
  any year at a station is 0.01, there should be on
  average 10 events larger than 20 m3/sec in 1000
  years, if the conditions affecting floods at the
  site do not change.
• The average recurrence interval between floods is
  100 years, so we refer to such a discharge as
  the 100-year flood..
• The floods will not occur regularly every 100
  years

99
Recurrence interval

• The probability of exceedence of the 100-year
  flood remains the same in any year
• It is 0.01 the year after a flood of this size
  occurred
• Probability that a discharge of this size will
  not occur in any year is (1-p)
• Probability that such a discharge will not be
  exceeded in N years is 1 pN
• Probability that such a discharge would be
  exceeded in n years is
• Best way to refer to a T-year flood is by means
  of its odds ratio --- it has a 1 in 100 chance
  of being exceeded in each year.

100
Regional flood-frequency curves

• Multiple-regression formulae based on data from
  all the USGS gauging stations in a region.
• Typical formula
•  
• where A drainage area
• Ei are watershed characteristics, such as mean
  annual precipitation, average elevation, average
  slope, etc.
• Obtained from US Geological Survey publications
  entitled Regional flood-frequency analysis for
  (State).
• Average estimation errors for the Potomac R.
  basin 20 for 2-yr flood 25 for 10-yr
  flood 40 for 50-yr flood.

101
Uncertainty in assessing flood risk

• Short records (Bulletin 17B suggests using at
  least 10 years of record!).
• There is no fundamentally representative
  theoretical probability distribution. Policy for
  using (say) Log Pearson Type III is based on the
  assessment that applying it to many flood records
  yields minimum standard errors of estimate. But
  reasons that are not understood physically.
• Best to try fitting more than one distribution
  and examining the uncertainty
• Methodological uncertainties (e.g. the outlier
  problem)

102
Uncertainty in assessing flood risk (contd.)

• Persistence problem
• Climate change
• Effects of dams --- confine analysis to post-dam
  period
• Watershed change --- e.g how to assess the
  influence of the non-steady expansion of logging
  through the Oregon Cascades?
• Some changes are reversible (e.g. canopy
  re-establishment)
• Others are not (e.g. many roads and ditches)