We have already seen that uncertainty about the future evolution of the climate is not your friend because it means things could be worse than anticipated. And we have shown that as uncertainty grows, then it is almost inevitable that the expected damage from climate change will also increase.buil

The reason that expected damage goes up together with uncertainty is because the overall relationship between global temperature rises and expected damages is non-linearly increasing: That is, a little bit of warming costs very little, but as temperatures continue to increase, the costs grow disproportionately. There is a figure in the previous post which visualizes this relationship.

How do we know that the “damage-function”, which relates temperature increases to damage, is non-linearly increasing in this manner? For one, virtually all economic models have converged on the same conclusion, namely, that increasingly greater warming accelerates the associated cost−thereby guaranteeing that greater uncertainty translates into greater risk, as we saw last time.

In this post, I show why greater uncertainty translates into greater cost in another way: Rather than relying on economic modeling, I use fairly simple mathematics. This has the advantage that we can avoid the complexity of economics and still make the point rather forcefully. So in the remainder of this post I explain recent work by Tasmanian scientist John Hunter that has explored the issue beautifully (Hunter, 2011).

Let us consider sea level rise (SLR) for this example because it has the advantage of being easily and unambiguously interpretable: We can all agree that being dry is good whereas being flooded is bad.

## Staying Dry

Suppose you have built your dream home on the coast of Patagonia within easy reach of your favourite beach. How likely is your home to get flooded? On average you will, of course, be dry−that’s the whole point of building a home safely above sea level as dictated by the rules of your local council or shire. Local governments have had plenty of experience to determine where it is safe (enough) to build.

But how did those council engineers make that determination? Roughly speaking, they combined knowledge about the average sea level with an *allowance *for extreme events. To clarify this distinction consider the top panel in the figure below.

The figure shows current average sea levels (indicated by the vertical blue line) together with the distribution of possible sea levels at any given instant. Due to ocean currents, tides, storms, and so on, there is natural variability in sea level all around the world, even without climate change. Hence you will experience sea levels that vary around the average (i.e. the mean) from day to day, as shown by the distribution in the top panel. It is therefore insufficient to look at the mean alone: We must make an allowance for variation as well.

That allowance is determined by policy makers to guard against all but the most extreme events. Thus, levees are built to withstand storm surges and your beach-front property must be built sufficiently far from the ocean so it does not get inundated easily. That precautionary allowance is labeled by the arrow in the top panel, which extends to the red line representing the maximally extreme events that building codes allow for.

However, very occasionally some outrageously extreme event, such as the conjunction of a high tide and a vicious storm, may cause your property to nonetheless be flooded. This would correspond to any sea level to the right of the red line in the top panel. Anyone who buys property close to the beach knows that such an extreme event might occur every so often, perhaps every 100 years or so.

Now enter climate change and the associated sea level rise (SLR).

The consequences of SLR are shown in the bottom panel of the preceding figure. It is apparent that mean sea level (the solid vertical blue line) will be greater than before. For simplicity, let’s suppose that the average SLR will be 50 cm by century’s end. (There is no need to get bogged down in the details, but that value is not an unreasonable assumption. So let’s use it in the remainder of this post without expressing a firm commitment to that particular value or any other.)

Now here is the interesting part: As a beachfront property owner, you aren’t just concerned about mean SLR by itself: instead, you are primarily concerned with *how often *your property will be flooded in the future. This is an important realization because it turns out that we need to be concerned with the additional allowance for extreme events on top of mean SLR.

How much do we need to allow for natural variation on top of mean SLR in order to keep the risk of flooding constant, on the order of 1 in a 100 years (or whatever they may have been before climate change came along)?

This total allowance is indicated by the red question mark in the bottom panel of the figure because we don’t know yet what that is.

Let’s find out.

## Keeping Floods at Bay

We begin by assuming that we want to keep the flooding risk to your property constant even after the sea has risen by .5 m. Obviously, we need to extend the height of our protective dam or levee (or raise our house onto stilts) by at least .5 m, just to cope with the average increase.

But that’s just the beginning.

Now we need to work out how much *extra *we have to allow to be prepared for extreme events−in other words, if the sea rises .5 m on average, how much will it rise when there is an extreme event, such as a storm tide? Or equivalently, by how much do I have to extend my levee or dam in order to keep the number of inundations constant? What is the required allowance, on top of mean SLR, to keep my risk unchanged?

Enter uncertainty.

## Allowances for Uncertainty

It turns out that this allowance for extreme events is a function of the uncertainty in the estimate of the mean expected SLR. If we take the mean and additionally estimate the uncertainty in that expectation, then we can compute the extra height of dams and levees that we need to build in order to allow for future extremes.

Hunter (2011) provides the mathematics that are necessary to compute these allowances, and I do not present them in detail here because this post is about the role of uncertainty not the mathematics per se. (If readers are interested in the details, leave a comment and I may add them later.)

The results of his analysis are shown in the figure below.

The large panel on the left shows the allowance that is required to cope with an SLR of .5 m as a function of the uncertainty in that estimate.

For now, we can ignore the differences between the three functions and consider them as one family of curves that show the same quantity: How much extra sea level do I have to allow for in order to keep my risk of flooding constant if on average I expect an SLR of .5 m.

Consider first the horizontal dashed line: That line represents the mean expected SLR, which is why it is drawn at .5 m. If there were no uncertainty in that estimate, then that’s all we had to allow for−which is why all three functions in the panel converge onto the dashed line on the left, as uncertainty (on the X-axis) shrinks towards zero.

If our knowledge were perfect, we’d allow for .5 SLR and we could relax (after putting our home on 50 cm tall stilts).

Now consider the implications of greater uncertainty: All three functions rapidly diverge from the dashed horizontal line as uncertainty increases and they do so in an accelerating manner. The greater the uncertainty in our projection of SLR, the greater the allowance we have to make in order to keep our exposure to floods constant.

Note that this is a result of greater uncertainty *only*−the average expected SLR remains constant; the only thing that changes is the precision of the estimate (expressed here as the standard deviation of the SLR projections).

This result derives directly from the mathematical properties of extreme values and is not in doubt: The greater the uncertainty, the greater the required allowance to protect against extreme events.

Now consider the three thumbnail panels on the right of the figure: They represent three different distributions of the expected SLR, with the familiar normal (bell-shaped) distribution in the top panel, and a raised cosine and a uniform (rectangular) distribution in the center and bottom thumbnails, respectively. The color coding of those distributions conforms to the color of the functions in the left-hand panel.

It can be seen that if uncertainty about SLR is normally distributed, increasing uncertainty has a particularly drastic effect (blue line in the main panel). For example, if uncertainty is .362 m (a value consonant with post-IPCC results; Nicholls et al., 2011), then under this assumption of normality one would have to allow for more than 1 meter of SLR to keep the risk of flooding constant−that’s double the expected SLR of .5 m! With the other two distributions, uncertainty has a lesser but still non-linearly increasing effect.

The fact that all three curves are non-linearly accelerating despite widely divergent assumptions about the distribution of uncertainty implies that the main result is extremely well supported. Greater uncertainty translates into a greater need to allow for variation on top of SLR, and that allowances grows in an accelerated fashion.

This result follows from basic mathematics, without any economic modeling, and it reveals particularly starkly why uncertainty is no one’s friend.

This concludes our three-part examination of the role of uncertainty in assessing the damage cost arising from climate change.

The next issue to be considered in future posts is how uncertainty affects the flipside of the equation; namely, the cost of mitigation. Irrespective of damage costs, does uncertainty make mitigation more imperative or does it lessen the urgency of action?

Take a guess….

## References

Hunter, J. (2011). A simple technique for estimating an allowance for uncertain sea-level rise*Climatic Change, *DOI 10.1007/s10584-011-0332-1.

Nicholls, R. J.; Marinova, N.; Lowe, J. A.; Brown, S.; Vellinga, P.; de Gusmão, D.; Hinkel, J. & Tol, R. S. J. (2011).

Sea-level rise and its possible impacts given a `beyond 4°C’ world in the twenty-first century*Philosophical Transactions of the Royal Society (A), **369*, 161-181