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Stage 4: Data analysis

Calculations: cross-sectional area | hydraulic radius | velocity | discharge | kinetic energy | Cailleux Index

Displaying your data: tables of results | plotting river variables on scattergraphs | displaying bedload data

Calculations

(i) Calculating cross-sectional area

Cross-sectional area = channel depth x channel width

If you have taken more than one measurement of channel depth you can calculate the mean.

Alternatively you can plot the width and depth readings on graph paper, then count the area of the stream. Maths types may wish to use the trapezium rule to calculate the area.

(ii) Calculating hydraulic radius

Hydraulic radius is calculated simply as follows

cross sectional area ÷ wetted perimeter

If you have measured bankfull width, depth and wetted perimeter, you can also calculate bankfull cross-sectional area and bankfull hydraulic radius, and compare these to the figures calculated above.

(iii) Calculating velocity

Find the mean time for the float to travel the set distance at each site. Velocity is then simply calculated as

velocity = distance ÷ time

For example, if a float travels 10m in 8 seconds,

velocity = 10 ÷ 8

velocity = 1.2 metres per second

Of course, even if you started the float off at different points across the width, it is likely to have been sucked into the line of fastest flow, probably in the centre of the stream. The results are, therefore, biased towards the fastest parts of the stream, and will not fairly indicate the velocity elsewhere. To overcome this bias, multiply the result that you have calculated by 0.8. Studies have shown that this produces a more realistic indicator of velocity across the whole width of the stream.

So for the example above where simple calculated velocity = 1.2 metres per second

1.2 x 0.8 = 0.96 metres per second

(iv) Calculating discharge

To work out discharge for any particular point you first need to have calculated mean velocity and cross sectional area.

Discharge = velocity x cross sectional area

If velocity = 0.96 m/s and cross-sectional area = 1.25 sq m

discharge = 0.96 x 1.25 = 1.2 cubic metres per second (or cumecs)

(v) Calculating kinetic energy

To work out the kinetic energy of the moving water at any particular point you first need to have calculated discharge and velocity.

Use the discharge figure to calculate mass of water per second. To do this, we use the fact that 1 cubic metre of absolutely pure water weighs 1 gram at 4 degrees Celsius. Therefore, x cubic metres of water will have a mass of x grams, e.g. 10 cubic metres weighs 10 grams.

Kinetic energy = (mass x velocity) / 2

If mass = 1.2g and velocity = 0.96 m/s

Kinetic energy = 1.148 joules per second

(vi) Calculating the Cailleux Index

Your raw data for each pebble is the radius of curvature (r) and the length of the A axis (l). For each stone, calculate Cailleux Index as follows Ci = (2r/l) x 1000.

A perfectly spherical pebble would have a Ci of 1000. The lower the Ci, the more angular the pebble.

Plot a frequency distribution of the Ci. Sort the results into a tally chart showing the frequency (i.e. number) of measurements in each 5 interval (0-5, 5.1-10, 10.1-15, etc).

Data presentation

Tables of results

Produce a summary table showing the results (and averages) for each sample site.

Plotting river variables on scattergraphs

The most straightforward scattergraph that you can plot is distance downstream on the x-axis against one river variable (width, wetted perimeter, cross-sectional area, hydraulic radius, mean depth, mean velocity, discharge) on the y-axis. Add a line of best-fit.

You can test the significance of the relationship between the two variables by calculating the Spearman's Rank Correlation Coefficient.

(iii) Displaying bedload data

If you have used Power's Index of pebble shape, you can display the data in proportional bar charts (see right) or pie charts to show downstream changes.

Data on bedload size (including the Cailleux Index) can be plotted on scattergraphs.

Statistical tests

You need to have collected data from at least 10 stations along the river. Calculate the Spearman's Rank Correlation Coefficient to test the significance of the relationship between distance from source and the variable (e.g. velocity, discharge, hydraulic radius, etc). Need more information about this test?

Worked example

The null hypothesis is that there is no significant correlation between cross-sectional area and gradient in Boggy Brook. The results were collected and summarised as follows.

Gradient	Cross-sectional area
x	y
0.018	0.70
0.037	0.54
0.059	0.43
0.081	0.97
0.045	1.08
0.025	1.01
0.016	0.98
0.016	0.86
0.007	2.05
0.017	1.78
0.006	2.55
0.021	2.01

Give each result a rank

Gradient		Cross-sectional area
x	rank	y	rank
0.018	6	0.70	3
0.037	9	0.54	2
0.059	11	0.43	1
0.081	12	0.97	5
0.045	10	1.08	8
0.025	8	1.01	7
0.016	3.5	0.98	6
0.016	3.5	0.86	4
0.007	2	2.05	11
0.017	5	1.78	9
0.006	1	2.55	12
0.021	7	2.01	10

Now calculate the difference between the ranks (d) and the square of each of those differences (d2).

Gradient		Cross-sectional area
x	rank	y	rank	d	d2
0.018	6	0.70	3	3	9
0.037	9	0.54	2	7	49
0.059	11	0.43	1	10	100
0.081	12	0.97	5	7	49
0.045	10	1.08	8	2	4
0.025	8	1.01	7	1	1
0.016	3.5	0.98	6	-2.5	6.25
0.016	3.5	0.86	4	-0.5	0.25
0.007	2	2.05	11	-9	81
0.017	5	1.78	9	-4	16
0.006	1	2.55	12	-11	121
0.021	7	2.01	10	3	9

Find the sum of all the squared differences (Σd2). Here it equals 445.5

Use this formula to calculate the Spearman's Rank Correlation Coefficient.

In this example, rs = -0.558

Now compare the calculated value of rs against the critical values for 12 pairs of measurements.

In our example, at the p=0.05 probability level, the critical value of rs = 0.587

Since 0.558 < 0.587 (ignore the - sign), the null hypothesis is accepted at the p=0.05 level

There is no correlation between gradient and cross-sectional area.

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