Principle of Index-Velocity Method and its Application Randy Marsden Teledyne RD Instruments

Principle of Index-Velocity Method and its Application

Randy Marsden

Teledyne RD Instruments

Summary

• Principles

• Example

• Practical Procedures

Part 1:

Index-Velocity Method: Principles

Why is an Index-Velocity method needed?

• Need: Continuous discharge measurement for open channels where simple methods like stage-discharge relationship do not give reliable results

• Examples:– Tidal rivers– Backwater conditions– Canals or rivers with control structures

• Establish a relationship between channel mean velocity and an Index-Velocity

• Index-velocity is a velocity measured at a local area (sampling volume) on the cross-section.

What is Index-Velocity Method?

• Developed by USGS in 1972• Used in U.S., China, France, Great Britain,

Japan, Canada, Mexico…….

• Instruments for Index-velocity Horizontal ADCP, i.e., ChannelMaster Acoustic travel time instruments

Index Velocity Method

In practice, three types of local velocity can be used as Index-velocity

• Horizontally averaged velocity at a depth

• Depth averaged velocity in a vertical

• Point velocity

Three Types of Index-Velocity

Point velocity

Depth averaged velocity

Horizontally averaged velocity

• H-ADCP (horizontally-looking) or travel time system. – Example: ChannelMaster

• Bottom-mounted ADCP: looking-up– Example: ADFM

• Point current meter for point velocity– Example: Marsh Mcbirney EM meter

How to measure Index-velocity?

Fundamentals

Discharge equation:

Q = A Vmean

Q = Discharge

A = Cross-section area

Vmean = Channel mean velocity

Cross-section area is a function of stage

A = f (H)

H = stage

A site may already have a table or curve for the stage-area relationship

Index Velocity Method - Area

Mean Velocity

• Vmean = k * Vindex

• k may depend on depth

– Usually not the case on irrigation canals since depth does not vary as much as natural streams

0

5

10

15

20

25

0 50 100 150 200 250 300 350 400

Station (feet)

Sta

ge

(fe

et)

Surveyed Standard

Channel needs to be surveyed for a selected “standard” cross-section to compute channel areas for a range of stages: stage-area ratingMan-made channels may use known dimensions.

Determining Cross-section Area

Channel area is always calculated at the “Standard” Cross-section

• H-ADCP not necessarily mounted at the “Standard” Cross-section location but it should not be too far away

Which cross-section?

A gauging station

1 = standard cross-section2 = wading measurement section3 = bridge measurement section

• Q1 = Q2 = Q3• Area is always computed at location 1!

1 2 3

GageCM

Stage-Area Rating

In many cases, stage-area rating may be expressed as:

A = a1 + a2 H + a3 H2

a1, a2, a3 = coefficients H = Stage

Rating curve = regression equation

One parameter regression

V = f (Vi )

Two parameter regression

V = f (Vi , H)

Index-Velocity Rating

A general, two parameter (Index-velocity and stage) linear regression:

Vmean = b1 + (b2+b3 H) VIndex

VI = Index-velocity

b1, b2, b3 = regression coefficients

Need at least six measurements at different velocities and depths to

due full regression

Linear Regression

One parameter linear regression

If b3 = 0:

Vmean = b1 + b2 VIndex

That is, channel mean velocity is a linear function of

Index-velocity.

Need at least four measurements at different velocities

and depths

Simple Linear

• If have only one or two measurements

Vmean = b2 VIndex

• Found to work well in canals and many rivers

when there is downstream control

Rating Development

Step One: Field data

collection

• Use ChannelMaster to

measure index velocity

• Use Rio Grande or

StreamPro to measure Q

and A

Rating Development

Step 2: Regression analysis

• Data collection need to be

conducted over a range of stage

or discharge to obtain a series of

data for Index-velocity and

channel velocity. Regression

analysis using a least-square

method to obtain Index-velocity

rating curve or equation.

• Same for area.

X Variable 1 Line Fit Plot

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2

CM VELX

ME

AN

CH

AN

NE

L V

EL

Y

Predicted Y

Rating Results

• Q = Vmean(Vindex, H) * A(H)

• Simplest Case: trapezoidal canal

• Q = k * Vindex * (a2H + a3H2)

• a2 is width of bottom of canal

• a3 is slope of canal banks

Accuracy

• Accuracy depends on quality of rating data.– How accurate and reliable is measured index

velocity?– How well does index velocity represent the

mean velocity – how good is k?– How accurate and reproducible is measured

discharge and area?

Canal 18

• Take data with ChannelMaster1200 kHz20 each 0.5 meter cells30 pings, 0.5 sec/ping

• Result for 35 minutes of dataVavg = 0.329 m/s ± 2.3%

Canal 18 Continued

• Rio Grande Discharge data

– 17 discharge measurements

– Vmean = 0.283 m/s ± 1.4%

– Amean = 41.06 m2 ± 1.7%

– Qmean = 11.63 m3/s ± 1.9%

Canal 18 Rating

• b2 = 0.86 ± 2.7%

• a2 = 8.01m ± 1%

• a3 = 2.87 ± 1%

• Q = (b2*VIndex)*(a2* H + a3 *H2) ± 3%

• This is for each 30 second discharge measurement.

Canal 18

• Since the discharge measurement noise is primarily random it could be reduced by doing more pings during the 30 seconds. By reducing the ping time to 0.1 seconds, and pinging for 20 out of 30 seconds, the noise of Vindex would be reduced to ± 1.3% and the uncertainty of the discharge to ± 2.0%.

How is this possible?

• The ChannelMaster and the Rio Grande both use BroadBand ADCP technology which gives:

• Low noise velocity measurement in short averaging times –

• a narrowband ADCP needs 50 times as many pings to reach the same precision for the same cell size.

BroadBand ADCP cont.

• BroadBand ADCPs can use smaller cells to measure the water

• For the ChannelMaster this means that there are more velocity measurements across the canal and they are closer to each bank – better accuracy for Vindex.

• For the Rio Grande this means more vertical depth cells with less estimated flow – better accuracy for Vmean.

Other reasons for BroadBand

• Less pings = less powerSmaller batteriesSmaller solar panels

• Pick the best number of cells and cell size for each siteCover more of flowReduce uncertainty

Part 2:

Application Example

Index-Velocity Rating Development at Imperial Irrigation District, California,

December, 2003

Imperial Irrigation District CaliforniaTrifolium 13 Check structure

600 kHz CM H-ADCPmounted upstream the check structure

Acousti cBeams

Mean FlowDi rectionX

Y

Z

Y

Cel l 1 Cel l jH-ADCP

H-ADCP

H0

0

CanalBank Canal

Bank

Canal Bottom

Water Surface

Sketch for ChannelMaster H-ADCP set-up

StreamPro ADCP used for discharge measurement

H-ADCP Parameter settings:

Cell size: 0.5 meterNumber of cells: 20Blank distance: 0.5 meterAveraging Interval: 37.4 secondsSampling Interval: 37.4 seconds

Screenshot from WinRiver software when

playing back a StreamPro data file

Time series of range averaged Vx for Cells 1 through 4 and

water level at the sampling/averaging interval of 37.4 seconds

0

0.1

0.2

0.3

0.4

0.5

0.6

12:00:00 13:12:00 14:24:00 15:36:00 16:48:00

Time

Ve

loc

ity

(m

/s)

0

0.1

0.2

0.3

0.4

0.5

0.6

Wat

er L

evel

(m

)

Velocity Water Level

Organizing Data for Regression Analysis

i

ki

ki

j

jjxkI VV

4 4

1, )(

4

1

5

1

Index-Velocity: calculate average velocity from CM during the time of a StreamPro velocity measurement

k = 1, 2, 3

Stage: Directly from H-ADCP vertical beam

)(]67.0

[ ADCPbottomADCP ZHWZH

A

Imperial Irrigation District Westside Highline CanalBed Geometry

0

1

2

3

4

5

0 3 6 9

Canal Boundary

Channel Master

Water level

compute from shape of canal

Note: H=1.07m, W=3.0 so

A=1.5Z2ADCP+6.21ZADCP +4.93

Cross-section Area

A

QV measuredmean

Canal Mean Velocity:

Get Qmeasured from StreamPro, Rio Grande, or ‘conventional” methods

Partial Data from StreamPro ADCP and ChannelMaster H-ADCP Organized for Index-Velocity Rating Development

StreamPro ADCP Measurement ChannelMaster H-ADCP Measurement

Transect Start Time

Measured Discharge (Qmeasured)

[m3/s]

Canal Mean

Velocity (Vmean)

[m/s]Sample

Start Time

Water Level (H)

[m]

Index-Velocity

(VI)

[m/s]

Cross-Section Area (A)

[m2]

12:44:56 2.482 0.304 12:44:56 0.470 0.351 8.175

12:49:03 2.264 0.280 12:49:18 0.460 0.336 8.098

12:57:01 1.914 0.239 12:57:24 0.453 0.274 8.041

13:01:31 1.391 0.172 13:01:46 0.455 0.199 8.060

13:11:05 0.954 0.120 13:11:07 0.435 0.146 7.909

13:14:41 0.783 0.099 13:14:51 0.425 0.127 7.834

13:21:01 0.574 0.074 13:21:05 0.413 0.088 7.740

13:24:57 0.474 0.061 13:24:49 0.405 0.069 7.684

13:36:20 0.256 0.034 13:36:03 0.385 0.045 7.536

13:40:36 0.247 0.033 13:40:24 0.388 0.045 7.555

Regression equation:

Vmean vs. VI

y = 0.8606x

R2 = 0.995

0.000

0.050

0.100

0.150

0.200

0.250

0.300

0.350

0.400

0.000 0.100 0.200 0.300 0.400 0.500

VI (m/s)

Vm

ean

(m

/s)

2 m range Linear (2 m range)

Imean VV 8606.0

A stage-discharge rating cannot be created at this site

0

0.5

1

1.5

2

2.5

3

3.5

0.300 0.350 0.400 0.450 0.500 0.550

Water Level (m)

Dis

char

ge

(m^

3/s)

Time series of rated discharges by applying the rating to the H-ADCP data and StreamPro ADCP measured

discharges on December 9, 2003

0

0.5

1

1.5

2

2.5

3

3.5

12:00:00 13:12:00 14:24:00 15:36:00 16:48:00

Time

Dis

ch

arg

e (

m3/s

)

SP Rated Q

Rating evaluation

Regression coefficient: R or R2

Standard Error

Used as indication of goodness of fit: closer to 1.00 is a better fit

Part 3

• Procedures and recommendations• Site selection• Mounting depth• Pitch and roll• Mount• Cell size• Selection of the good cells

Site Selection

• Choose site with best aspect ratio

• Aspect ratio is width/center depth

• Do not want beam hitting bottom sooner than necessary

Mounting depth

• Mount at 50-60% of mean low water elevation. This is near the average velocity point of the vertical profile.

• Provides widest range of operation

Pitch and Roll

• Mount with pitch and roll as close to zero as possible– Maximizes useful range– Beams looking at same plane of the water– Requires pitch/roll sensor

• Use the Mount ADCP screen in WinHADCP to assist setting up.

• After maintenance you can be sure that CM is pointed in the same direction to prevent a shift of the rating.

The Mount

• The mount should be:

– Rigid: shaking can introduce unwanted noise– Adjustable: to allow pitch and roll to be set

close to zero– Retractable: to allow routine cleaning– Reset easily: to put ADCP back to original

orientation

Example mounts

Cell size

• Select a good cell size for the application• Compromise between low noise and maximum

profiling range• Large cells have low noise but may limit how close

you can get to the far bank – small cells averaged together have same noise as one large cell of the same width.

• 20 cells is enough for most applications• SDI-12: up to 27 cells for SDI-12 Version 1.2 and

twenty cells for Version 1.3

Selection of good cells

• Do not want to use cells contaminated by far bank.

• Look at intensity and correlation plots do determine maximum useful profiling range

• Intensity appears to show data ok to 8.8 meters

• But we see that that cell ‘looks wrong’

Selection of good cells continued

Correlation data shows that cell at 8.8 meters has correlation contamination and should not be used

?Questions?

LinearCurvilinear Compound

One Parameter Rating Forms

One Parameter Curvilinear Ratings

x

y

Polynomial

y = b + c1x + c2x2 + c3x3...

x

y

Logarithmic

y =c1ln(x) + b

x

y

Exponential

y =c1ebx

Power law

y =c1xb

x

y

One Parameter Compound Ratings

A B

Vi

V A

BTransition

A = Linear

B = Linear