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Welcome! My name is Hubert Savenije and I
am a hydrologist.

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The most visible and most intriguing part
of the hydrological cycle is the runoff.

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Every person has at some stage looked in fascination
at flowing water, or been inspired by the

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beauty or violence of a stream

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Flowing water seems alive. And it is an inseparable
part of our life.

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Water has been essential to create and support
our societies

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and sometime it threatens it with its destructive
force

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As we saw earlier, it took philosophers and
scientists some time before they realised

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that river runoff is completely dependent
on precipitation

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At first they thought that there was not enough
precipitation to feed the rivers, but in fact

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there is a lot more than that.

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Globally, less than 40% of the precipitation
leads to runoff.

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As precipitated moisture travels through the
terrestrial cycle, it is stepwise partitioned

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into other fluxes.

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The first partitioning point is the surface.

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If rain falls on water or on saturated land,
then it runs off directly to the open water.

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We call this saturation excess overland flow.

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Above land, rainfall is intercepted by:

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leaves, the litter on the ground and the ground
itself (where it may pond)

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and from there it evaporates back to the atmosphere.

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The typical time scale of this process is
between an hour and a day.

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If the ponding is large enough, water may
flow overland and find its way to the stream.

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We call that infiltration excess overland
flow, or Hortonian overland flow, after the

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great American hydrologist Horton

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This overland flow is very fast, with a time
scale of hours.

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The remainder infiltrates into the soil, from
where it can transpire or percolate into the

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groundwater.

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During a rainfall event, the soil can become
so saturated that pockets of saturation occur

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under the ground at relatively shallow depth.

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If the terrain is sloped, then this sub-surface
pocket may connect and generate subsurface

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runoff through a network of preferential channels.

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These channels have been created by the water
itself, making use of root channels, animal

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burrows and fissures.

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In mountainous and hilly area, this is a dominant
mechanism, which we call storage excess subsurface

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flow or shortly ‘interflow’

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The water that percolates to the groundwater
subsequently feeds the very slow groundwater

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flow, that sustains the base flow of the river.

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Finally, groundwater can be exchanged with
neighbouring areas or may rise to the unsaturated

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soil by capillary rise.

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The many storages and thresholds in this system
are the reason why the rainfall-runoff processes

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is highly non-linear,

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although the individual processes can often
very well be described by linear processes,

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just like we saw with the groundwater seepage.

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So the fast surface runoff is the precipitation
minus the interception and the infiltration

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The fast sub-surface runoff is the infiltration
minus the transpiration, the soil evaporation

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and the percolation

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And the base flow equals the percolation minus
the capillary rise

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It is hard to distinguish these different
runoff components once the water has entered

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the river.

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But the water can reveal its origin when we
look at its colour and composition.

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If it is very turbid, and flowing fast, then
it stems from surface runoff.

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If it flows fast and is mostly clear, then
it stems from rapid subsurface flow.

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If the water is clear and the flow is quiet,
then it consists of seepage flow from GW

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From the chemical elements in the water, the
isotope composition and the turbidity we can

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deduct the origin of the water,

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but also the time it spent in the catchment
before it came to runoff

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But that goes beyond this introductory course.

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Depending on the climate and the physical
properties of the landscapes in the catchment,

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rivers have different runoff signatures

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Here we see clearly different signatures from
two neighbouring catchments

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On average, the Rhine has about ten times
more discharge than the Meuse

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Not only is the runoff of the Rhine larger,
it is also less variable over the year

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What makes these patterns so different?

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Is it climate?

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“It’s the landscape, the climate is not
so different”

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The mountainous part of the Rhine, with its
snow cover and glaciers makes that the runoff

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during the dry summer months is still substantial,
whereas the runoff of the Meuse in de dry

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summer months is very low.

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But climate can be very important.

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Whether a river is ephemeral, intermittent
or perennial depends primarily on the climate

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Traditionally we measure discharge by a current
meter at a certain position in the cross-section

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of a river

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We find the total discharge by integration
of all these points over the cross-section

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This is a lot of work, although it is fun

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As you can see

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But although its is great fun doing it in
this way, it is also a bit old fashioned.

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Nowadays we use Doppler instruments in the
stream that are able to map the entire velocity field

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or we make use of movies from iPhones to infer
the discharge from the flow pattern observed

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from the top

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But this is something for an advanced course

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We use discharge measurements to compose rating
curves, linking the water level to the discharge

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The relation between depth h and discharge
Q is a power function which plots a straight

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line on double log paper provided we have
the zero of the reading right: the so-called

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h0, or the gauge reading when the discharge
is zero

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These rating curves are expressions of the
Manning formula for ‘uniform’ flow.

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The power function of the rating curve completely
matches the Manning formula below.

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The Manning formula consists of two parts:
the first part is a constant for the given

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section and the second part depends on the
water level h

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The width, the depth and the hydraulic radius
R all depend on the depth, whereas the roughness

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coefficient n and the slope S does not

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Interestingly, the water level dependent part
purely depends on the geometry of the cross-section

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Stevens developed a very smart way of combining
the two.

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The first part (the curve), one can derive
from a topographical survey of the cross-section.

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This curve can be extrapolated as far as necessary,
as high as the water level may go.

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The other part, the line, required a couple
of discharge measurements,

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but because it is a straight line, it requires
only a limited number of points

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And subsequently we can use that line to combine
with the curve

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Have a look at the example to see how it works.

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Rating curves can have peculiarities, On double
log paper, they should plot on a straight line,

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(if you select the right H0)

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but they can show kinks (inflections) in the
curves, which can bend up as well as down.

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Here you see a hand-drawn rating curve made
in the 1980’s

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of the Limpopo river in Mozambique.

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The dots with the years were observations
during big floods, when the whole of the flood

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plain of the river was covered with water.

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The inflection point is at the water level
where the river starts to fill the floodplain

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and where the flood plain offers an additional
and much wider channel

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This is Bangladesh, the delta of the Ganges,
Bramaputhra and Meghna.

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Look at the upper Eastern corner, called Sylhet

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This is where world record rain is falling

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During the monsoon the Sylhet area is deeply
flooded, up to more than 4 m deep.

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You might think this is because of the heavy
rainfall in the area, but it is because of

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the backwater from the Meghna, where it joins
the Ganges and the Bramaputhra

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Here you see this backwater effect.

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During the pre-monsoon, in May, the floods
in the Surma, which drains Sylhet, can discharge

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uninhibited to the Gulf of Bengal,

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But during the peak of the monsoon, the discharge
of the Ganges and Bramaputhra is so high,

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that it causes a great backwater.

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This causes rating curves to tilt up

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We see that these backwater curves are labelled
as design floods with a probability of

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once in a 100 and once in 20 years.

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How do we determine flood levels with these
probabilities?

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In the chapter on precipitation, we prepared
duration curves to rank the rainfall events

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according to their magnitude,

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In that way we obtained frequency curves.

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For extreme floods, we shall make use of Gumbel's
theory for annual extremes.

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Here you see how a hydrograph can be turned
into a duration curve, showing the % of time

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that a certain discharge is exceeded, or in
this case ‘not-exceeded’.

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We call this the probability of non-exceedence

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The interesting part of this curve is the
upper part,

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and moreover, we would like to extrapolate
the curve to rare probabilities such as once

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in a 100 years: a probability of non-exceedence
of 99%.

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Gumbel developed a statistical distribution
for such extremes.

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His statistics are useful for determining
design discharges or extreme flood levels

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such as these

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The basic assumption underlying Gumbel’s
theory is that the underlying phenomenon

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(in this case river flow, or alternatively
water levels or precipitation) are normally

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distributed

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But this is unfortunately seldom true 

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additionally the data series needs to be homogeneous
(caused by a single population of events)

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It should also be stationary (no climatic
change or man-induced changes)

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and long enough (a reasonable part of the
return period)

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But also these things are seldom true

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Is Gumbel’s theory then still useful?

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Well, yes, but you have to use it with good
common sense

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These are the equations.

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I am sure they look difficult and complicated
to you,

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but in fact they are not.

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They are much easier than the normal distribution,
which is very similar

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Gumbel’s equation for q (the probability
of non-exceedence) is simpler than Gauss’

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equation, which we see here.

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Unlike the normal distribution, Gumbels's
equation can be calculated easily with a pocket

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calculator or an iPhone

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Moreover, the reduced variate looks similar

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To carry out a Gumbel analysis, you have to
select the extremes of a series.

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Normally we look at annual extremes.

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Say you have N years of data, then you take
the N highest values of these years

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You then rank these extremes in order of magnitude

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To determine their frequency, you use the
plotting position

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To prevent that the lowest value in the series
gets a probability of 1, you add one to the

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number of years in the denominator

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As a result, the highest value in 9 years,
has a probability of exceedence of once in

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10 years

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Here is a Table with highest peaks over a
period of 7 years.

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A short series, merely as an example.

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We select the 7 highest values and rank them

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The highest value then has the probability
of exceedence of once in 8 years.

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Gumbel paper is prepared in a way that the
horizontal axis for the reduced variate y

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is linear.

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The Gumbel equation then plots as a straight
line against the vertical ordinate.

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The probability of non-exceedence belonging
to these y values are calculated with the

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Gumbel formula resulting in the typical probability
pattern of the graph.

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The vertical ordinate is the x, which has
a linear relationship with y

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I’ll show you an example of a Gumbel line
for Thailand

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where as you can see severe floods can take
place such as in 2011, when all the bank were

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overtopped

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I'll show you the Gumbel graph of Phraya Banlu

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It clearly shows that the highest values bend
of when the banks overtop

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Just like we saw happening with the rating
curve

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Of course the kink disappears when you build
a dike on both sides of the river

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But what is the probability that a flood with
a return period of T actually occurs within

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a period of n years? It's this !

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Try it yourself in the exercise

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In applying Gumbel’s theory, you should
always remain critical

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Series are seldom normally distributed

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You cannot extrapolate the theory much beyond
the length of the series and you must always

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be aware that series are not homogeneous.

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Moreover, people are always tampering with
the hydrological system

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and as a result, its characteristics change
over time

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This was maybe not an easy part of the course,
but if you study it closely, and if you practice,

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then it is simpler than it seems.

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In the next module we'll discuss the runoff
generation mechanisms.