Flood Recurrence Intervals and Probabilities

In this week’s lab, we’re going to learn how
we use records of streamflow to determine the expected discharge of a 100-year flood;
that flood that has a 1% chance of being seen, or in any given year. Again, it’s not a prediction of when flooding
occurs, but rather, what is the probability of seeing a flood of a given size. The graph that you’re looking at shows peak
annual discharges for the Willamette River at the Albany measuring station. What’s plotted here is the highest flow of
discharge for each year, and it’s just simply flooded plotted as a time series. One of the things that you immediately notice
about this graph is that there’s been a change in the size of the peak annual discharges. Back in this period, say before the mid 1960s,
we see larger peak annual discharges and then the peak annual discharges abruptly drop after
about the mid-1960s. This is reflecting the influence of dam construction
on the upper Willamette and it’s tributaries. The dams serve a variety of purposes, but
one of those purposes is to hold back water during periods of high flow, thereby reducing
the peak flow events. In other words, reducing the size of floods
on the lower river. The dams provide a storage capacity to hold
water back upstream, and then release it more slowly over time, thereby reducing the peak
flow because there’s an obvious change in the size of the peak annual discharges. The frequency of discharges of different sizes
has obviously changed over the time period of record. Larger flows were more frequent back in the
earlier time period and then become less frequent as we move to the present. So for this reason, we’re going to do the
same analysis twice for the Willamette River at Albany. I’ve picked a 10-year period here, 1950 to
1960, and another period of equal length 1970, 1980. as a result of the analysis, we’ll see
how the probability of seeing discharges different sizes and the magnitude of those discharges
has changed with time as a result of changes in the watershed. Dam construction is an example of one change,
but land-use changes and their effect on runoff, potential climate changes, and so forth can
all affect the size of discharges and frequency or probability of seeing a discharge of different
sizes. This table shows the peak annual discharges
for specific years. Again, I’m going to do the analysis twice,
ones for each of these time periods so that we can see how the results change, how the
probability floods of different sizes has changed as a result of dam construction on
the Willamette River. The first step in this kind of analysis is
to rank the peak annual discharges from 1 being the largest, two being the second-largest,
all the way down to n, where n is the number of years in the data set. each of these data sets has 11 years, and
so the smallest flood is going to have a rank of 11. Here I’ve gone through and ranked the discharges
in each data set by size. Largest discharges ranked number one in each
data set, second largest discharge is ranked number two in each data set, and so forth,
all the way down to the smallest discharge, which is ranked 11th because each of these
data sets covers 11 years. The next step is to calculate the recurrence
interval for each rank. Recurrence interval is an expression of the
frequency, how often on average, we should expect to see a flood of that size given the
range of discharges we have in each data set. Recurrence interval is calculated simply from
the rank of that discharge. The formula for calculating each recurrence
interval is given here and is the number of years in the data set. In this case, N always equal, always equals
11 because we have eleven years of data in each set. M is the rank of a particular discharge; largest
discharge is equal to 1, the smallest discharge is equal to N or has a rank of n, and in our
case being 11 because we have 11 years of data in each set. For example, the discharge rank number one
has a recurrence interval equal to 11 plus 1 divided by 1; 11 is n, 1 is M or rank, and
n–number of years in the data set. So our largest flood rank equals one as a
recurrence interval of 12 years in each data set. When we actually plot our data, we’re going
to be plotting recurrence interval versus discharge. Recurrence interval is going to be the x-axis,
discharge is going to be the y-axis, and as a result, we’re actually going to get two
different graphs. As you can see there’s a different discharge
associated with each recurrence interval because the discharges in the data sets are different. the flood ranked number two is going to have
recurrence intervals six years. Again, 11 is n, 2 is the rank, so 11 plus
1 divided by 2 gives us 6 years. The second largest flood has a recurrence
interval of six years with n, number of years of data equaling 11. the discharge that’s ranked number three as
a recurrence interval of four years in each data set–again because the number of years
of data is 11 in both cases and so on. Calculate recurrence interval for each rank
using n plus 1 divided by M, and is the number of years of data, or to the number of years
in the data set, and M is the rank of that flood or discharge. The next step is make one graph for each data
set. In other words, make a separate graph for
each data set. Your recurrence intervals are similar, but
the discharges associated with each of those recurrence intervals changes. Recurrence interval is going to be on the
x-axis, discharge is going to be on the y-axis. In other words, x equals the recurrence interval,
y equals the discharge for that recurrence interval. Now these graphs may look a little bit different
than what you’ve seen before because the x-axis is a log scale; by that I mean it’s powers
of 10. The y-axis is a linear scale which is the
normal scale that you’ve seen many times before. When you’re making your graphs, you’re going
to have to choose a scale for your y-axis based on the range of discharges you see in
your data set. for example, the Willamette river dataset covering the years 1950, 1960
has discharges ranging from 174,000 CFS cubic feet per second to 62 thousand cubic feet
per second. So we’re gonna scale our y-axis from about
oh 60,000 CFO’s to about 180,000 CFO’s. So again, we don’t want our data to fill any
more than about half of the y-axis and you’ll see why in a minute. Ultimately what we’re going to do is after
plotting our data, we’re going to extend the relationship that we observe between recurrence
interval and discharge out to the 100-year recurrence interval. We need to leave ourselves room to do that
because the discharge for the 100 year recurrence interval is likely to be much larger than
the range of discharges in our data set. So if you don’t leave yourself room on your
y-axis to extend your line to the hundred year discharge, it’s going to go off the top
of the page, and you’re gonna have to redo your graph from the beginning. Once we’ve got our data points plotted, 11
data points for each graph, we’re going to draw a line through that data set. We’re not going to connect the dots but rather
we’re gonna draw one straight line, a best-fit line, and draw it in such a way that there’s
an equal number of points above and below that line. The way a computer would draw a best-fit line
is it would calculate the total distances of points above and below the line. In other words, the sum of the distances above
the line would be equal to the sum of the distances of points below the line. We’re then going to extend our line, one single
straight line all the way out to the 100-year recurrence interval. So here’s the graph we’re gonna plot X, X
this is recurrence interval; units of years. Y-axis is discharge; units of cubic feet per
second. Let’s plot the Willamette River, 1950 to 1960
dataset first. Range of values in that set go from a high
of about 174 thousand cubic feet per second, to a low about 62 thousand cubic feet per
second. As a result, I’ve scaled my y-axis from 60
to 180,00 CFO’s to incorporate the range of data in that set, and I’ve scaled the y-axis
in such a way that I’m not filling more than about half of the y-axis, leaving myself room
to extend my line all the way out to the 100-year discharge without going off the top of the
page. So the flood with recurrence interval 12 in
the 1950 to 1960 data set, has a discharge of a hundred and seventy four thousand cubic
feet per second. So it’s gonna plot about right there. the flood with a recurrence interval of six
years has a discharge of 155,000 CFO’s plotting about right there. The flood ranked three recurrence interval
of four, its discharge is 146,000 CFS plotting around there and so on. Flood ranked four; recurrence interval equals
three years; discharge 112 thousand CFO’s. flood ranked five; recurrence interval of
two 2.4, 2.1, 2.2, 2.3, 2.4 as a recurrence interval of a hundred and six thousand CFS. Recurrence interval two years; discharge 99,000
CFO’s. recurrence interval 1.7 years; discharge 82,000 CFO’s. Recurrence interval 1.5–one, two, three,
four, five; that’s 1.5, its recurrence interval or its discharge, eighty thousand CFS. Recurrence interval 1.3; discharge seventy-seven
thousand CFS. Recurrence interval one 1.2; discharge 63,000
CFS. currents interval 1.1; discharge 62,000 CFS. We’ve got our eleven data points plotted. I’m now going to lay a ruler, adjust the fit
of my line, and draw a line again that doesn’t necessarily pass through all the points, but
rather results in an equal number of points above and below the line, and we’re going
to draw that line all the way out to the 100-year recurrence interval. What’s the discharge associated with the one
year recurrence interval? Up to 10, to 40, to 70 that; would be 300,000
CFS. we’re sitting at about two 285,000 cubic feet per second. In other words, the 100-year flood, the flood
with a recurrence interval of 100 years or a one percent chance of occurring in a given
year, has an expected discharge of 285,000 cubic feet per second. You would do the same thing for each of the
data sets because the range of discharges is different in each data set. We’re gonna get a different expected discharge
for the 100 year flood. Here we’ve plotted the data for the second
data set, Willamette River, 1970 to 1980. We’re now going to draw a best-fit straight
line like we did before and find out what the expected discharge is for a 100-year flood. Notice that one of the data points looks like
an outlier at data point for the 1.1 recurrence interval, it’s quite a bit lower than anything
else. So in drawing our best fit line, we’re going
to largely identity–ignore the data point that we’ve identified as an outlier. I’m going to position our line in such a way
that we have an equal number of points above and below the line, and we’re going to draw
it all the way out to the 100 year recurrence interval and then read off the expected discharge–expected
discharges about a hundred and sixty-five thousand cubic feet per second for the 100
year recurrence interval. It’s quite a bit different than what we found
in the previous dataset, reflecting what we expected to see from the graph that we started
off with. We saw the peak discharges have decreased
after the mid-1960s with the flood control capacity provided by the completion of dams
in the upper part of the Willamette watershed. So what does all this mean? What we’re getting again is not a prediction
of when the flood is going to occur, but rather we’re learning what to expect in terms of
the size of a given flood. What we’ve learned is the expected discharge
for a flood of the 100 year recurrence interval. We can get the expected discharge for a flood
of any recurrence interval we might be interested in, but typically the 100-year recurrence
interval is what’s used because that serves as the legal definition of areas that are
at risk of flooding. Now why don’t here–100-year flood does not
mean that a flood of that size would occur once every 100 years. Instead what it means, or it’s better interpreted
as a probability, a chance. Probability is just the reciprocal of the
recurrence interval. I one under your flood–has a 1% chance of
occurring in any given year. They meant–that might not sound like much,
but risk is cumulative. Each year, you have one percent chance of
seeing a discharge of that size we expect based on the analysis we just did. Let’s say you take out a 30-year mortgage
to buy a house, within the lifespan of your mortgage there then becomes a 30% chance of
seeing at least one 100-year flood. Oh it’s 1% times 30 years, we have a 30% chance
of seeing at least one, one under your flood within the life of that mortgage, now it’s
starting to sound a bit riskier. This is why banks will not lend you the money
to buy that house unless you also carry flood insurance on that house, because banks are
not willing to take that risk. You’re free to take that risk but the bank
is not willing to do so without the flood insurance protection. So in mapping out areas that are at risk of
flooding, we would take the discharge that we expect to see well for a 100-year flood
as given by the analysis we just did, convert that to a height or elevation, then map out
that elevation on a topographic map, showing the area that would be flooded when the river
flow is that high. If your property maps within the area flooded
at the 100 year discharge, you live in that rivers legal floodplain, and if you have a
mortgage you’ll be required to pay flood insurance premiums as a condition of that mortgage.