We have measured sea level at the Battery in New York for over 150 years, from 1856 to the present, albeit with a 14-year gap from 1879 to 1893. The monthly-average sea level data are available online from NOAA (as are the data from hundreds of tide gauge stations around the world). They even provide a convenient graph:
If all you did was look at that graph naively, with no analysis of the data behind it, you might be tempted to think that “Sea levels have been rising at a fairly steady pace since at least the mid-1800s.” But if you study the graph, and the data behind it, you find that’s not true at all.
It shows the data itself (the jagged line in blue) together with the “best-fit” straight line (the one which matches the data better than any other straight line). As such, it estimates the straight-line trend and its rate of sea level rise, which is a good estimate of the average rate since 1856. In this case, it’s rising at 2.88 mm/yr.
But — if the rate of sea level rise has changed (something we call “acceleration”), then the average rate since 1856 is not the same as the rate now.
Here’s my graph of the same data:
We can simplify our graphs and our analysis, without losing information about the long-term trend, simply by working with yearly average sea level rather than monthly average sea level. That looks like this:
The slope of the best-fit straight line is only an estimate of the rate of sea level rise at NY, but a reasonably precise one: my analysis says the average rate since 1856 has been 2.88 mm/yr, within a “margin of error” of 0.16 mm/yr, meaning it (95% probability) isn’t less than 2.72 or more than 3.04. If you’re curious why my “margin of error” is bigger than that given by NOAA, see the note at the end of this post.
What if we looked at only the recent data, since 1990? We could fit a straight line to estimate its trend, and the margin of error for our estimate, and it looks like this:
The estimated rate is now 4.41 mm/yr, with a margin of error 1.38. It’s (probably) somewhere between 3.03 and 5.79. That’s a pretty wide range, but it definitely doesn’t include 2.88 — the overall average. This window on the data suggests that recently, the rate of sea level rise has been faster than its overall average — at least, at New York.
If we do the same to the 30 years before that, we get this:
This rate estimate is a mere 1.67 with margin of error 1.32, so it too has a wide range of uncertainty (smaller time spans generally do), but it’s still (probably) not less than 0.35 or more than 2.99. It could be 2.88 as the entire time span suggests — that’s on the high end but still plausible — but it’s definitely nowhere near the 4.41 we saw in the most recent period. The difference is considerable; according to these data, over the last 32 years the tide has risen at NY 2.6 times as fast as it did the previous 30 years. It’s starting to look like the rate of sea level rise at New York has not been steady.
I can do the same for segments of 30 years or so over the entire observed time span:
It’s definitely starting to look like the rate of sea level rise at NY has not been steady.
You might be wondering, why did I pick 30-year time spans? Why start the latest one at 1990? Did I try lots of possibilities, and pick the one which most makes it look like the rate of sea level has changed?
That’s a good question, and a very important statistical point. Allowing yourself to make that choice, dramatically increases your chance to find something that “looks like” it’s significant, but is really just a random accident due to random fluctuations. It’s like buying a lot of lottery tickets instead of just one: the lottery is still random (at least, we hope it is) but you have more chance to “win” by accident because you took more chances. It’s called the “multiple testing” problem, also known as “selection bias.”
That’s why we apply very stringent tests to models that allow that kind of choice, before we declare they are “statistically significant” with any degree of confidence.
In this case, the “set of straight lines over segments starting at 1856, 1900, 1930, 1960, and 1990” model passes those tests. Yes, the rate of sea level rise at NY has changed over time.
Although the “five straight lines” model is better than the “one straight line” model (the one with constant rate of sea level rise), and the improvement is “statistically significant” (it passes those tests), an even better model — both statistically and physically — is five straight lines which meet at their endpoints.
Such a model (straight line segments meeting at their end-points) is a good general-purpose model which allows for the rate to change, and is sometimes called a PLF, or “Piece-wise Linear Fit.” The straight-line model all by itself imposes constant rate as a constraint; the PLF retains the simplicity of the straight line but accomodates changing rates.
The PLF allows the rate to change, but only at the “breakpoints” between segments. Hence just as the single-straight-line model estimates the average rate (and its margin of error) during the whole time span, each segment of the PLF estimates the average rate (and its margin of error) during each time segment.
I’ll graph those estimates as a solid blue line showing the rate during each time segment, together with light blue shading to show its uncertainty range:
The thin dashed line at 2.88 mm/yr shows the single-straight-line rate, but it’s clear that the actual rate has usually been significantly higher or lower than that. There are two episodes of pronounced sea level rise, from 1930 to 1960 and from 1990 to the present. The current rate is highest of all.
There are other kinds of models which allow for a changing rate, and aren’t tied to specific time spans. One of the workhorses of my toolkit it the lowess smooth, which I’ve programmed to compute both the rate of change and its uncertainty (margin of error), and when I use it on the data from NY (the Battery) it suggests that sea level has changed thus (in red, compared to the PLF estimate in blue)
Both methods reveal that there are two episodes of faster-than-average sea level rise at NY, and that the latest is going on right now. They both suggest that the current level of sea level rise at NY is higher than it has been before. And yes, their results are “statistically significant.”
The data from NY (the Battery) are an excellent introduction to what sea level has been doing, especially along the east coast of the U.S., and how sea level rise has gotten faster, then slower, and now is faster than ever. A similar pattern is present in lots of tide gauge records (but not all), and also reveals itself in global mean sea level; that too shows a complex pattern of change in the rate of sea level rise.
Those changes are easy to see in global sea level since the year 1900, but not so easy to see in data from a single location (like NY). We had to do more than just “look at a graph,” we even did some actual analysis. Still, it was possible because NY covers such a long time span and gives us so much data to work with. Not every tide gauge record yields such results, because of the high noise level in single-station records. The global mean has a much lower noise level, making rate changes easier to quantify.
A high noise level always makes rate changes hard to confirm and even harder to see. That’s one of the reasons, I think, that those who want to dismiss or minimize the danger of sea level rise, so often will show a single graph of a single tide gauge station (often the Battery in New York) and declare that sea level rise has been steady. They wish to imply that it will remain so in the future, and that the rate right now is no more than the long-term average. They can’t deny that sea level is rising, that’s too much nonsense for anybody, but by plotting a single graph of a single station with a straight line already on it, you can get — and they can give — the wrong impression.
The tide gauge station at the Battery in New York is a prime example. It’s also a prime example of the fact that if you look closely, the “steady rate” story starts to fall apart, and if you do the math, the “steady rate” story crumbles.
Who would do that, show a single tide gauge record and then declare “steady rise” with no analysis behind it? The so-called “Heartland Institute,” for one. They created an entire document targeting (according to them) teachers and students, purporting to give “facts” about climate change but actually designed to give the wrong impression. When it comes to sea level rise, they show a single tide gauge record, do no analysis at all, and declare (direct quote) “Sea levels have been rising at a fairly steady pace since at least the mid-1800s.” The record they chose is: the Battery in New York.
Judith Curry, for another, when she talked about how sea level rise will impact New Jersey from “a business perspective.” She showed a single tide gauge record, did no analysis at all, and said (quote) “Since 1910, sea level has been rising at a steady rate of 1.36 feet, or 16 inches, per century.” At least she chose the data from Atlantic City, NJ.
Rutgers University put a lot of effort and expertise into advising the state of New Jersey about dealing with future sea level rise. Judith Curry’s advice is to pay no heed to their estimates of how much sea level rise New Jersey will have to deal with in the near future. Woe betide the garden state, if they do as she suggests.
That’s why I call the “Heartland Institute,” and Judith Curry, deniers.
[Note: NOAA puts a “+/- 0.09” after their trend estimate, while I put a “+/- 0.16”. That number is the estimated uncertainty in the trend rate. The estimate is based on the behavior of the noise, the random fluctuations in the data. We both recognize that the noise isn’t “white noise” (the simplest kind). They use an AR(1) model for the noise, I use an ARMA(1,1) model, and my model usually allows more influence from the noise so it returns larger margins of error.
The salient point is that because of that, my calculations make it harder for tests of rate change to reach “statistical significance.” But they reached it anyway.]
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