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Statistical and practical challenges in

estimating flows in rivers

- From discharge measurements to hydrological

models

Motivation

- River hydrology Management of fresh water

resources - Decision-making concerning flood risk and drought
- River hydrology gt How much water is flowing

through the rivers? - Key definition discharge, Q
- Volume of water passing through a
- cross-section of the river each time
- unit.
- Hydraulics Mechanical properties

of liquids. Assessing discharge under

given physical circumstances.

Key problem

- Wish Discharge for any river location and for

any point in time. - Reality No discharge for any location or any

point in time. - From B to A
- Discharges estimated from detailed measurements

for specific locations and times. - Simultaneous measurements of discharge and a

related quantity gt relationship. Time series of

related quantity gt discharge time series. - Completion, ice effects.
- Derived river flow quantities.
- Discharge in unmeasured locations.

2000

3/3-1908 now

3/3-1908 1/1-2001

3/3-1908, 12/2-1912 13/2-1912 ..

Annual mean, 10 year flood

1980

1/3-2000, 23/5-2000 14/12-2000 5/4-2004

1/1-2000 now

Annual mean, Daily 25 and 75 quantile, 10 year

flood, 10 year drought

1960

1940

22/11-1910, 27/3-1939 5/2-1972 8/8-2004

27/3-1910 now

1920

100 year flood

1/8-1972 31/12-1974

15/8-1972, 18/4-1973 31/10-1973 .

1900

1) Discharge measurements and hydraulic

uncertainties

- Discharge estimates are often made using

hydraulic knowledge and a numerical combination

of several basic measurements. - De-composition of estimation errors
- Systematic contributions method, instrument,

person. - Individual contributions.

1) Discharge measurement techniques

- Many different methods for doing measurements

that results in a discharge estimate (Herschy

(1995)) - Velocity-area methods
- Dilution methods
- Slope-area methods

1) Velocity-area methods

- Basic idea Discharge can be de-composed into

small discharge contributions throughout the

cross-section. - Q(x,y)v(x,y)?x?y

x x?x

y

y y?y

A

x

1) Velocity-area measurements

- Measure depth and velocity at several locations

in a cross-section. Estimate

Lambie (1978), ISO 748/3

(1997), Herschy (2002).

Alternative Acoustic velocity-area methods (ADCP)

Current meter approach

L2

L1

L4

L3

L5

L6

v1,1

v5,1

v2,1

v4,1

v3,1

v1,2

v5,2

v4,2

d1

d5

v2,2

v3,2

d4

d2

d3

1) Current-meter discharge estimation

- Now Numeric integration/hydraulic theory for

mean velocity in each vertical. Numeric

integration for each vertical contribution.

Uncertainty by std. dev. tables. ISO 748/3 (1997) - Could have Spatial statistical method

incorporating hydraulic knowledge.

Calibration errors number of rotations per

minute vs velocity. Creates

dependencies between measurements done with the

same instrument.

v

v8

v1

v7

v4

v2

v9

d1

v6

d7

v3

d2

d6

v5

d3

d5

rpm

d4

1) Dilution methods

- Release a chemical or radioactive tracer in the

river. Relative concentrations downstream tells

about the water flow. - For dilution of single volume QV/I, where V is

the released volume and I is the total relative

concentration, - and rc(t) is the relative concentration of

the tracer downstream at time t. - Measure the downstream

relative

concentrations as

a time series.

1) Dilution methods - challenges

- Uncertainty treated only through standard error

from tables or experience. ISO 9555 (1994), Day

(1976). - Concentration as a process? Uncertainty of the

integral. - Calibration errors. (Salt temperature-conductivit

y-concentration calibration)

t

1) Slope-area methods

- Relationship between discharge, slope, perimeter

geometry and roughness for a given water level. - Artificial discharge measurements for

circumstances without proper discharge

measurements. - Mannings formula Q(h)(A(h)/P(h))2/3S1/2 /n,

where h is the

height of the water surface, S is the slope, A is

the cross-section area, P is the wetted perimeter

length and n is Mannings roughness coefficient.

Barnes Davidian (1978) - Area and perimeter length geometric

measurements.

P(h)length of A(h)Area of

h

1) Slope-area challenges

- Current practice Uncertainty through standard

deviations (tables) ISO 1070 (1992). - Challenge Statistical method for estimating

discharge given perimeter data knowledge about

Mannings n. - Handle the estimation uncertainty and the

dependency between slope-area measurements.

1) General discharge measurement challenges

- Ideally, find f(e1, e2,,en s1, s2,,sn,C,S),

ei(Qmeas-Qreal)/Qreal, sispecific

data for measurement i, Ccalibration data,

Sknowledge of other systematic error

contributions. - User friendliness in statistical hydraulic

analysis. - What we have got now

f(e1, e2,,en )fe(e1)fe(e2)fe(en)

2) Making discharge time series

- Discharge generally expensive to measure.
- Need to find a relationship between discharge and

something we can measure as a time series. - Time series of related quantity relationship to

discharge - Discharge time series
- Most used related quantity Stage

(height of the water surface).

2) Water level and stage-discharge

- Stage, h The height of the water surface at a

site in a river.

Stage-discharge rating-curve

h

Q

h0

Datum, height0

Discharge, Q

2) Stage time series stage-discharge

relationship discharge time series

h

Q

Maybe the stage series itself is uncertain, too?

2) Basic properties of a stage-discharge

relationship

- Simple physical attributes
- Q0 for h?h0
- Q(h2)gtQ(h1) for h2gth1gth0
- Parametric form suggested by hydraulics (Lambie

(1978) and ISO 1100/2 (1998)) QC(h-h0)b - Alternatives
- Using slope-area or more detailed hydraulic

modelling directly. - Qab h c h2 Yevjevich (1972),

Clarke (1994) - Fenton (2001)
- Neural net relationship. Supharatid (2003),

Bhattacharya

Solomatine (2005) - Support Vector Machines. Sivapragasam Muttil

(2005)

2) Segmentation in stage-discharge

- QC(h-h0)b may be a bit too simple for some

cases. - Parameters may be fixed only in stage intervals

segmentation.

h

h

width

Q

2) Fitting QC(h-h0)b, the old ways

- Observation QC(h-h0)b q?log(Q)ab

log(h-h0) - Measure/guess h0. Fit a line manually on

log-log-paper. - Measure/guess h0. Linear regression on qi vs

log(hi-h0). - Plot qi vs log(hi-h0) for some plausible values

of h0. Choose the h0 that makes the plot look

linear. - Draw a smooth curve, fetch 3 points and calculate

h0 from that. Herschy (1995) - For a host of plausible value of h0, do linear

regression. Choose h0 with least RSS. - Max likelihood on qiab log(hi-h0) ?i ,

i?1,,n, ?i N(0,?2) i.i.d.

2) Statistical challenges met for QC(h-h0)b

- Statistical model, classical estimation and

asymptotic uncertainty studied by Venetis (1970).

Model qiab log(hi-h0) ?i , i?1,,n, ?i

N(0,?2) i.i.d. Problems discussed in Reitan

Petersen-Øverleir (2006) - Alternate models Petersen-Øverleir (2004),

Moyeeda Clark (2005). - Using hydraulic knowledge - Bayesian studies

Moyeeda Clark (2005) and Árnason (2005), Reitan

Petersen-Øverleir (2008a). - Segmented curves Petersen-Øverleir Reitan

(2005b), Reitan Petersen-Øverleir

(2008b). - Measures for curve quality curve uncertainty,

trend analysis of residuals and outlier

detection Reitan Petersen-Øverleir (2008b).

2) Challenges in error modelling

- Venetis (1970) model qiab log(hi-h0) ?i ,

?i N(0,?2) can be written as QiQ(hi)Ei,

EilogN(0,?2), Q(h)C(h-h0)b. - For some datasets, the relative errors does not

look normally distributed and/or having the same

error size for all discharges?

Heteroscedasticity.

Residuals (estimated ?is) for segmented

analysis of station Øyreselv, 1928-1967

2) More about challenges in error modelling

- With uncertainty analysis from section 1

completed - Uncertainty of individual measurements and of

systematic errors. - With the information we have
- Modelling heteroscedasticity. So far, additive

models. Multiplicative error model preferable. - Modelling systematic errors (small effects?).
- Uncertainty in stage gt heteroscedasticity?
- ISO form not be perfect gt model small-scale

deviations from the curve? Ingimarsson et. al

(2008) - Non-normal noise / outlier detection?

Denison et. al (2002)

2) Other QC(h-h0)b fitting challenges

- Ensure positive b.
- Not really a regression setting stage-discharge

co-variation model? - Handling quality issues during fitting rather

than after (different time periods). - Handling slope-area data.
- Doing all these things in reasonable time.

Prioritising

Before flood After

flood

2) Fitting discharge to other quantities than

single stage

- Time dependency changes in stage-discharge

relationship can be smooth rather than abrupt.

Can also explain heteroscedasticity. - Dealing with hysteresis stage time

derivative of stage. Fread (1975),

Petersen-Øverleir (2006) - Backwater effects stage-fall-discharge model.

El-Jabi et. al (1992), Herschy (1995),

Supharatid (2003), Bhattacharya Solomatine

(2005) - Index velocity method - stage-velocity-discharge

model. Simpson Bland (2000)

3) Completion

- Hydrological measuring stations may be

inoperative for some time periods. Need to fill

the missing data. - Currently Linear regression on neighbouring

discharge time series. - Problem
- Time dependency means that the uncertainty

inference from linear regression will be wrong.

3) Completion meeting the challenge

- Challenge Take the time-dependency into account

and handle uncertainty concerning the filling of

missing data realistically. - Kalman smoother
- Other types of time-series models
- Rainfall-runoff models
- Ice effects Ice affects the stage-discharge

relationship. Completion or tilting the series to

go through some winter measurements? Morse

Hicks (2005) - Coarse time resolution -

Also

completion?

3) Rainfall-runoff models (lumped)

- Physical models of the hydrological cycle above a

given point in the river. Lumped works on

spatially averaged quantities. - Quantities of interest precipitation,

evaporation, storage potential and storage

mechanism in surface, soil, groundwater, lakes,

marshes, vegetation. - Highly non-linear inference. First OLS-optimized.

Statistical treatment Clark (1973). Bayesian

analysis Kuczera (1983)

P

E

T

S0

S1

S5

S4

S2

S3

Q

4) Derived river flow quantities

- Discharge time series used for calculating

derived quantities. - Examples mean daily discharge, total water

volume for each year, expected total water volume

per year, monthly 25 and 75 quantiles, the

10-year drought, the 100-year flood.

4) Flood frequency analysis

- T-year-flood QT is a T-year flood if

Qmaxyearly maximum

discharge. - Traditional Have
- Sources of uncertainty
- samples variability Coles Tawn (1996), Parent

Bernier (2003) - stage-discharge errors Clarke (1999)

- stage time series errors Petersen-Øverleir

Reitan (2005a) - completion
- non-stationarity

5) Filling out unmeasured areas

- For derived quantities regression on catchment

characteristics - Upstream/downstream scale discharge series
- Routing though lakes.
- Distributed rainfall-runoff models. Example

gridded HBV. Beldring et. al (2003)

From an internal NVE presentation by Stein

Beldring.

Layers

Derived quantities in unmeasured areas

Discharge series in unmeasured areas

Meteorological estimates

Hydrological parameters

Derived quantities

Stage time series

Parameters inferred from discharge sample

Completion

Rating curve

Individual discharge measurements

Model deviances

Other systematic factors

Instrument calibration

Conclusions

- Plenty of challenges. Not only statistical but in

the possibility of doing realistic statistical

analysis information flow. - Awareness of uncertainty in the basic data is

often lacking in the higher level analysis.

Building up the foundation. - User friendly combinations of statistics and

programming. - How much is too much?
- Computer resources
- Programming resources
- ISO requirements difficult to change the

procedures. - Sharing of research, resources and code.

References

- Árnason S (2005), Estimating nonlinear

hydrological rating curves and discharge using

the Bayesian approach. Masters Degree, Faculty of

Engineering, University of Iceland - Barnes HH, Davidian J (1978), Indirect Methods.

Hydrometry Principles and Practices, first

edition, edited by Herschy RW, John Wiley Sons,

UK - Beldring S, Engeland K, Roald LA, Sælthun NR,

Voksø A (2003), Estimation of parameters in a

distributed precipitation-runoff model for

Norway. Hydrol Earth System Sci, 7(3) 304-316 - Bhattacharya B, Solomatine DP (2005), Neural

networks and M5 model trees in modelling water

level-discharge relationship, Neurocomputing, 63

381-396 - Coles SG, Tawn JA (1996), Bayesian analysis of

extreme rainfall data. Appl Stat, 45(4) 463-478 - Clarke RT (1973), A review of some mathematical

models used in hydrology, with observations on

their calibration and use. J Hydrol, 191-20 - Clarke RT (1994), Statistical modeling in

hydrology. Wiley, Chichester - Clarke RT (1999), Uncertainty in the estimation

of mean annual flood due to rating curve

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gauging. J Hydrol, 31 293-306 - Denison DGT, Holmes CC, Mallick BK, Smith AFM

(2002), Bayesian Methods for Nonlinear

Classification and regression. John Wiley and

Sons, New York - El-Jabi N, Wakim G, Sarraf S (1992),

Stage-discharge relationship in tidal rivers. J.

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Representation and Approximation. Conference on

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References

- Fread DL (1975), Computation of stage-discharge

relationships affected by unsteady flow. Water

Res Bull, 11-2 213-228 - Herschy RW (1995), Streamflow Measurement, 2nd

edition. Chapman Hall, London - Herschy RW (2002), The uncertainty in a current

meter measurement. Flow measurement and

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Snorrason A (2008), Bayesian estimation of

discharge rating curves. XXV Nordic Hydrological

Conference, pp. 308-317. Nordic Association for

Hydrology. Reykjavik, August 11-13, 2008. - ISO 748/3 (1997), Measurement of liquid flow in

open channels Velocity-area methods, Geneva - ISO 1070/2 (1992), Liquid flow measurement in

open channels Slope-area method, Geneva - ISO 1100/2. (1998), Stage-discharge Relation,

Geneva - ISO 9555/1 (1994), Measurement of liquid flow in

open channels Tracer dilution methods for the

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1151-1162 - Lambie JC (1978), Measurement of flow -

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methods for fitting rating curves, with case

studies. Adv Water Res, 288807-818 - Parent E, Bernier J (2003), Bayesian POT

modelling for historical data. J Hydrol, 274

95-108

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Uncertainty in flood discharges from urban and

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311 188-201 - Petersen-Øverleir A (2006), Modelling

stage-discharge relationships affected by

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power-law regression with a location parameter,

with applications for construction of discharge

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