# on streamflow and sediment transport

**Topics …**

Annual Exceedance Probability (AEP)

**On Sediment Transport – Lane**

**(`I love it!’)**

Lane (citation) gave us a fantastic, and fantastically simple, `equation’ (relationship) for sediment transport in a river. It goes something like this: river flow (quantity, Q) and stream slope (or steepness, S) … go like … sediments size (diameter, D) and the amount of that size sediment that the river carries (Qs). In `equation’ form …

Q • S ~ Ds • Qs.

Note that I didn’t put an equal sign in there. It’s because it’s not an equality, at least in that form. For example, I suspect that the slope drives the relationship more to the ½ power. And there are other factors not included, like the shape and roughness of the river channel. But let’s look at what we have … I love how it explains everyday things, and maybe some not-so-everyday. It says that if one thing (Q, S, Ds, or Qs) changes, at least one other thing must also change, to keep the relationship in balance.

**Situation 1 – ****E****asy to ****U****nderstand.** Let’s say we have everything equal in a stream except for the flow quantity, Q (called `discharge’ among hydrologists, engineers). Imagine some river in its natural state, undamaged by man’s adventures; let’s say it’s `out in the wild’. We’re in the Southeastern USA, but the steam has not been tinkered with too much (by Man). It’s been a while since it rained; the stream is a trickle of clear (drinkable, or not; probably not) water. A huge rainstorm hits the area, and what happens? The discharge (flow rate) increases … and what else? You guessed it! … the stream becomes cloudy, silty. The increase in Q on the left side of the equation is reflected by an increase (or `start’) in the amount of transport, Qs, of sediment of size, Ds.

Silts (small-diameter sediment) get suspended in the flow as Q increases `a little’, and the stream is cloudy. Less visible there’s perhaps sand (bigger size Ds) bouncing along the bottom, and maybe small pebbles (even bigger Ds). So, as goes the quantity of flow, so goes the quantity, and size, of sediment (silt, sand, pebbles, whatever).

And what happens once the stream flow drops off? Yeah, sediments of various sizes can no longer be carried at their former rates, or not at all; they fall from suspension; the stream clears up. The more flow, the more sediment transport; slow flow is clear; flood flows are muddy. Easy, every-day stuff.

**Situation 2 – ****A****lso ****E****asy to ****U****nderstand, but ****M****aybe ****L****ess ****T****hought-of.** Let’s talk about Spring runoff from snow melt. For example, the Lochsa River in Northern Idaho, United States. Late Summer, Fall, and even into Winter, the river water is beautifully clear. You can look into the deeper pools and see enormous fish. Winter is the `wet’ time of year for the Lochsa, and the `wet’ comes in the form of snow, at least in the upper reaches. For much of the Lochsa drainage (the area that the Lochsa `drains’), the snow just accumulates; the stream flow doesn’t necessarily change that much. Then comes Spring; the temperatures warm, and Spring rains come. The snow just melts and is absorbed into the snow below, along with the rain. This is called `ripening’ of the snowpack. (Yeah, the entire region is covered with snow for a good fraction of the year.) Then, as we hit about June, the snow gets `ripe’; the warmer temperatures melt even more snow; the snow can absorb no more, and it, along with any rain, now go, into the stream. The flow quantity is now ten, twenty, maybe fifty times as great as the peaceful flow last September. What does the Lane relationship above say? Yeah, rocks the size of cars (big Ds!) are transported (bounce along downstream). (In fact, from time to time cars themselves run off the side of the road, into the River, and if it’s June, they bounce along with the rocks.) By late August the big rocks, and little ones, and the pebbles, and the sand, and probably even the silts (except for occasional rains, or landslides) have settled to the bottom and found their new resting places. (Presumably the cars have been located and towed.) So, as goes the flow quantity, so goes the *size* of the sediment transported.

But before we move to the next `Situation’, note that Qs and Ds are `on the same side of the equation’. What this means is that, numerically speaking, there is a trade-off between the two. It actually makes total sense. For a given amount of flow discharge, and stream slope (and shape/geometry, etc.), more of the little sediments are carried compared to the bigger ones (rocks, boulders, etc.). Any single combination of Q and S results in a combination of various Ds being carried along at Qs, though for some size Ds the flow rate is zero; the transport of sediment is not limited to a single size, Ds. And, of course, as the discharge, Q, drops off, first the larger sediments aren’t carried at all, and settle to the bottom. As the discharge drops off more, the next size smaller settles, and so on. Eventually the stream goes back to the relative trickle, and is clear!

**Situation 3 – Dams and Deltas! ** Yeah, now let’s say we have a stream, and Man comes along and builds a (damn) dam. There are several ways we can look at this `situation’. One way would be to say that the dam `stops’ the river. But with most dams the `stop’ is only temporary, and the water comes out of (or over) the dam, in overall the same flow rate as it comes in (except for maybe some evaporation, diversion for irrigation, etc.). So, instead of saying a dam `stops’ the water, let’s look at the water `slope’ … water *surface* slope. The water surface slope goes to (near) zero behind the dam. What happens? For *any* Q on the left side of the `equation’ (short term or long), the zero slope makes the entire left hand size zero … meaning we must also get zero on the other (right) side of the equation. Whoa, yeah; basically no sediment gets transported. A delta! (Well, maybe some of the really fine stuff keeps going, since our slope doesn’t go to exactly zero, and, even so, it may take a bit of time for the really fine stuff to settle out.) What do we get? The lake (reservoir) behind the dam begins to fill up with sediment. (But you know this. Or if you didn’t, you do now.)

**Situation 4 – Below the (Damn) Dam. ** This is where it gets fun. (Or dangerous!) The Lane relationship holds true at an instant of time, and also *over* time (days, months, years, decades). The river reaches a certain `equilibrium’ month after month, year after year, decade after decade; in high flows sediment is picked up and transported; in low flows it settles out. The river looks basically the same year after year, but for some overall erosion. But if we put in a (damn) dam, the river gets messed up. The river bed flooded over by the reservoir behind the dam gets covered with more and more sediment. The water exiting the dam is `clear’, or relatively so, even in *high* flows. (Perhaps you can see where I’m going with this.) Below the dam the river was `used to’ carrying various amounts of sediment associated with various amounts of discharge. But if the water coming out of the dam is clear (no sediments; it all got deposited in the reservoir), Lane’s relationship still demands that the river carry sediment! Where does it get the sediment? Yeah, the stream bed below the dam. The stream bed that was in happy equilibrium, since the dawn of time, until the (damn) dam got put in, is now out of balance. Instead of being happy with the sediment coming down from above, the river sucks it off the river bottom below the dam; the river bed scours. The channel deepens, gets carried away. This can be disastrous as it can undermine the dam and cause it to fail.

**Situation 5 – Smaller Streams. **Lane’s relationship was purported for large streams (rivers). But the same holds true for smaller streams (creeks). If you change one thing, you will necessarily change another, or others. If you build an embankment over a stream, with a culvert at the bottom, and if the embankment ponds a bunch of water behind it, the resulting pool will capture the sediment. And the stream below the embankment, now starved for sediment, will take it from the creek below the embankment – and there goes your embankment (and whatever was on top of it) – not to mention the flood of water and debris heading downstream.

**Situation 6 – Braided Streams. ** Sky Top Creek in the Beartooth Mountains in Montana is an example. Sky Top Creek drains Sky Top Lakes at the foot of Granite Peak, Montana’s highest. The Creek tumbles from the shelf holding the Lakes down to the elevations below. By “tumbles” I mean … cascades. The water is `white’ as the tumbling (high speed from the steep slope) causes air-entrainment (bubbles). Then the Creek hits a relatively flat spot, like a meadow. (Maybe the meadow was another lake at some point in time.) What happens? Kinda like with the dam! The sediment carried by the fast flow falls out … the meadow (or, first, lake) fills with sediment; it clogs. The stream is constantly clogged, blocked, by the sudden settling of sediment, damming it and damming it more. The stream (un-)braids, splitting up, trying to go this way, and that way, around the clogs. Both the cascading and the braided portions have their beauty. The cascading stream is picking up and carrying sediment, rocks, boulders; the braided stream is where they are dropping and clogging the stream.

**Situation 7 – Meandering Streams. ** This is pretty cool. Go to Google Earth, or some other map or map platform, and look at meandering rivers in, say, the middle of the Country. Pick a stream that is on a State (or County) line (boundary). You’ll see the meandering stream, and a meandering State (or County) line … *but they are different meanders!* What happened? Well, the line got surveyed, presumably down the river middle, or `thalweg’(?), and over time the river *moved*. Can Lane’s relationship explain this? Think about it.

And I wonder what the farmer thinks when he finds that part of the land he inherited, while still in the same state, is now on the other side of the river!

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### Meandering Streams

**What Causes a Stream or River to `Meander’?**

**(First) Hydraulic Equilibrium. **Let’s think about a river that flows into the ocean. But the same could be said for a river entering a lake, or some other feature that `fixes’ the eventual, or ending, river water level. The river dynamics are `controlled’ by the ending water level, the ocean. We could also say that the ocean `stops’ the river. And now let’s go upstream, up the valley containing the meandering river, perhaps the head of the valley, or a `great lake’, where the water level is, also, more or less, fixed. A meandering river is formed when the `straight line’ between these two points (upper end of the valley, and ocean) is too steep. The river would be constantly gaining speed. The river has to have some means of slowing down, so it meanders, e.g., `takes the long way’ (winds back and forth) down the valley. The fixed elevation difference and a longer flow path results in a lesser slope. Hydraulic equilibrium is reached.

**Sediment Transport Equilibrium.** And erosion and sediment transport still take place … some parts of the meandering river channel erode, especially at high flows, and deposits are made in other parts, and at low flow. And, overall, sediment coming into the meandering rivers stretch is transported through, and on out. The river, in hydraulic (and sediment transport) equilibrium, is constantly changing. Banks are eroded through and the river overflows or flows through to join the rest of the river somewhere downstream. Individual meanders get abandoned, causing `ox-bow’ lakes. The lakes may become stagnant, fill with lily pads, homes to noisy frogs, etc.

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**Statistics and Probability … **

### The 100-Year Flood

Streamflow statistics are interesting. Fascinating. I love streamflow stuff. Rivers, creeks, floods, storms, etc. Let’s start our talk with the `100-Year Flood’. Loosely, we say that it’s the flood that occurs `once in a hundred years’. It’s `The Big 0ne’! Happens once in a century. Huge amount of water. `Once in a lifetime’. And let’s say we have been keeping track of the floods, year after year, in a certain river. If we have been keeping track for 100 years, it would be that `doozie!’ … that flood that occurred just once; top of the pack; highest on the list of flows, or yearly floods. In one hundred years it occurred *once*. Lesser floods occurred more often. The next lesser flow occurred twice (the lesser flood plus the doozie). And so forth.

In terms of modern hydrology, this doozie has a `Return Period’ or `Return Interval’ of … 100 years.

And here’s where we take an important step, maybe a `leap’, in tackling streamflow, floods, hydrology, etc., *statistically, probabilistically*. Making predictions. We know what happened in the previous 100 years; what will happen in the future, the next 100? We say that the 100-year flood has a probability of `one in a hundred’ in any one (particular) year. On January 1^{st} we can say that the chance of experiencing the 100-year flood this year is … 1/100 … 1 percent. What is the chance of experiencing it the *next *year? Same; one in a hundred. What is the chance we will experience the 100-year flood *both* years? Let’s think about it. The chance in one year is small. The chance for the next year is small. The chance for both to occur is *really* small. In terms of probability it is … 0.01 x 0.01 = 0.0001 … one in 100 x 100 or 10,000 … or 1/100th of one percent.

Let’s ask the question … what is the probability that we will get a 100-year flood this year, or the next, or both? We answer this question (most easilty) by looking at the opposite. The opposite of the 100-year flood happening at least once this year and next is … that the 100-year flood occur *not at all* during these two years. The chance that the 100-year flood will `not’ occur this year (or *any* year) is … 99% … pretty high. Same with next year. But for the 100-year flood to not occur *both *years is … 0.99 x 0.99 = 0.98 … 98 percent chance. So the chance that we’ll get the 100-year flood this year or next (or both) is … the opposite (of the opposite) … 100 – 98 = 2%. One in 50. Kinda makes sense.

Let’s be bold and attempt to state the probability that we will have at least one 100-year flood in the *next* 100 years. We know it occurred exactly once in the last 100 years; what about the next 100 (assuming `no climate change’, etc.)?

From above, it is the opposite of no 100-year flood occurring for 100 consecutive years. This `non-occurrence’ probability is … 0.99 (year 1) x 0.99 (year 2) x … 0.99 (for year 100) = (0.99)^{100} = … 0.33 … 33 percent. The opposite of this is 100 – 33 = 67 percent. Huh? Yeah! It is *not* guaranteed that we will get a 100-year flood in the next 100 years … but we could say that it is `more probable than not’ (`two to one’ odds). Okay, what if we say the next 150 years? Well, let’s look … 1 – (0.99)^{150} = 0.78 … 78 percent chance … four to one … not a guarantee, but more likely than at 100. Try 500 years! … 1 – (0.99)^{100} = 0.99. So, yeah, finally … 99 percent chance … 99 to 1 … that we’ll get the 100-year flood at least once in a 500-year period.

There’s no guarantee that we’ll get a 100-year flood in the next 100 years. Or 50-year flood in the next 50.

Maybe a better question is … what is the chance we’ll get a 100-year flood in the next 30 years (the term of my mortgage)? Okay; here it is … 1 – (0.99)^{30} = 0.26 … 26 percent. There’s about a one-in-four chance we’ll see that flood (at least once) in the next 30 years.

And all of this depends on the next years following the behavior of the previous (historical/recorded) ones. (At some point climate change surely does kick in!)

** … Annual Floods and Mean An****nual Flood (MAF) …**

Let’s look at the floods that occur *each* year, or at least most years. We can think of these floods as those that essentially fill the banks of the stream or river (`bankfull’ discharge or flow) and to some extent overflow the banks, or `flood’. For example, for a flood with a 2-year occurrence interval we might say that it occurs once every two years, or `every other year’. More correctly we should say that this flood has a probability of 1 / 2 = 0.50 or 50% chance of occurring *each* year (and the next year, and so on). What is the probability that we will get the `2-year’ flood at least once in two years (this year and the next)? Similar to our examination of the bigger floods … we look at the probability of the 2-year flood *not* occurring in two years (`non-occurrence’). If the flood has a 50 percent chance of occurring in a year, it has a 100 – 50 = 50 percent chance of not occurring in the same year. The chance of it not occurring, not even once (not at all) in two years is 0.50 x 0.50 = 0.25 … 25 percent chance. Thus the chance of it *occurring* at least once in these two years is 100 – 25 = 75%! It’s not a done deal, but more probable than not! … three-to-one odds (75 versus 25) we’ll get the 2-year flood (at least once) in 2 years. Let’s try 5 years, just for kicks and giggles; 1 – (0.50)^{5} = 0.97 … 97%. Still not a guarantee, but pretty strong odds! But to be clear! … there is no guarantee that you will get the 2-year flood in two years … or the 5-year flood in the next five, and so on. If you really want to dial in and determine a flow that will occur at least every other year, you essentially need to look at a flow that occurs *every* year (return interval of 1). If we want to be 97% sure we get at least one 2-year flood in two years, we are `looking at’ a flow associated with a return interval of … 1.2, or 14.4 months. Of course floods don’t cycle like that, but if we put all of our flood flows to a curve (probability or return interval versus flood flow), then the flow associated with a return interval of 1.2, or probability of 1/1.2 = 83% (for any single year), would have a probability of 97% of occurring at least once in two years.

There is this thing called the Mean Annual Flood (MAF). In real life it is the arithmetic mean (average) of each year’s flood. Someone named Gumbel apparently was able to associate the mean annual flood with the flood that has a return interval of 2.33 years. I’m not quite sure how. And I’m not sure if this is an exact relationship, average relationship, a relationship for a certain type of stream, or just what. And this gets kind of weird since we don’t really ever look at streamflow chunks in 2.33-year (28-month) periods, or any other phenomenon (that I know of) in 2.33-year chunks. It is said, further, that the MAF has a probability of occurrence each year of 0.43 or 43 percent. (Yeah … 1 / 2.33 = 0.43; I get that; it’s just algebra.) That’s a bit more understandable. We could say that in any given year there is a 43% chance we’ll get a flood that is equal to or greater than the average of all the years’ floods. So? It’s just a *number*. It’s derived from a `curve’ (curve fit) that is developed from all the annual floods, and we would assume it is interpolated between 2 and 3.

What is the probability the MAF will occur this next year? … Answer: 0.43 or 43% chance. Almost 50-50.

What is the probability the MAF will occur next year? … Answer: 43%.

What is the probability the MAF will occur this year or next? … Answer: two ways to go about this …

First let’s look at the probability of the MAF not occurring in any given year … 1 – 0.43 = 0.57 … the probability of `nonoccurrence’.

So, the MAF occurring this year or next … can happen this way …

It occurs this year and not next … 0.43 x 0.57 = 0.24499, plus …

… not this year but yes next year … 0.57 x 0.43 = 0.24499, plus ..

or it occurs *both* years … 0.43 x 0.43 = 0.18419 … add up to …

0.67 … or 67% chance.

There’s a 67 percent chance (2 to 1 odds) that the MAF will occur this year or next (or both).

The other way to calc it is like this … what is the chance that the MAF will NOT occur both years?

0.57 x 0.57 = 0.3258, or 33% chance.

The opposite of that (the non-occurrence of the nonoccurrence) is 1 – 0.33 = 0.67 = 67%. (Same)

So, there’s a pretty good chance we’ll get the MAF at least once this year or next, but it’s not a `done deal’.

**Bankfull Discharge**

The flow rate that `just fills’ a stream or river banks is the so-called `bankfull discharge’, and has been found to be the discharge associated with a return interval between 1 and 2, or maybe more. We could say, then, that the river fills it banks more often than not … just about every year!

### Annual Exceedance Probability (AEP)

Okay, so, we have been talking about the 100-year flood, the Mean Annual Flood (MAF), etc. When we talk about the 100-year flood, we’re talking about a flood (flow) event associated with a return interval of 100 years (`100-year event’). In this case it would be a lot of water … `The Big One’. It would be a certain amount of water, either measured, or predicted (or estimated). Let’s say, purely for example, it’s 32,800 cubic feet per second (cfs, or `second-feet’) of water. This flood flow amount, as we `move forward’, e.g, … in the future, has a probability of occurrence (each year), of, 1/100 = 0.01 = 1%. But what it precisely means is that *that *particular flow amount has a probability of *being equaled or exceeded* of 1/100 or 1%. Thus the idea of `annual exceedance probability’ (AEP).

Similarly, when we talk about the probability of occurrence of, say, the MAF, we say it has a probability of occurrence of … 1/2.33 = 0.43 or 43%, where 2.33 is the `recurrence interval’ of the MAF, in this case 2.33 years. Let’s say the MAF (`number’) is 995 cfs. We’re not saying that there is a 43% chance that a flow of *exactly* 995 cfs will occur (any given year), but that a flow equal to or *exceeding *995 cfs will occur. Thus, `annual’ `exceedance’ `probability’. (There is essentially zero chance of the annual flood being exactly (only) 995 cfs in any year, or even in the next 10 years, or 50. And the chance of `exactly’ gets smaller and smaller the more `exact’ we’re talking about.)