Boston Sea Level

How one plots the data can have a big impact on whether or not one can “see” acceleration or deceleration of sea level in tide gauge data. Here, for instance, is a plot of sea level at Boston (data from PSMSL, and I’ve removed the annual cycle):

It’s similar to (and on the same scale as) the plot provided by NOAA.

One can easily get the impression the data follow a straight line (plus noise of course). One reason is that the y-axis scale covers 1.2 meters while the data only cover about 0.5 meters. Another reason is that the plot includes a straight-line fit, which to most people is highly suggestive of a straight-line pattern.

NOAA has good reasons for those choices. They want to plot all tide gauge records on the same scale, so they need a wide-range y-axis to accomodate them all. They also want to report the linear trend rate (whether the trend is linear or not), although as I’ve said before they make no attempt (and no pretense) to detect or to rule out departures from linearity. For their purpose, their scale choices and inclusion of the straight-line fit make good sense.

But for “seeing” acceleration and deceleration, this is a better plot:

It’s exactly the same data, but this plot makes it a lot harder to say “See — no acceleration.”

Despite the effect plot choices have on what we think we “see,” the real test is analysis. In fact I’ve often emphasized that we too often think we see meaningful patterns when it’s really just noise — that’s not the Mona Lisa next to the sistine chapel, it’s really a bunch of clouds that look like it.

I’ll start by computing annual averages, which reduces autocorrelation without sacrificing the precision of trend analysis. The annual averages are still autocorrelated, so I’ll still inlude an autocorrelation correction in analysis, but the annual averages make things a bit easier to “see.” Here they are for Boston:

Now it’s even harder to say “See — no acceleration.”

Nobody disputes that it’s rising. But is it really doing something else? It “looks like” there was deceleration early and acceleration later. Let’s try fitting a cubic curve, in order to accomodate such a pattern. It looks like this:

More to the point, the cubic term is statistically significant. Strongly. So no, it’s not following a straight line. According to this model it shows deceleration early and acceleration later.

That model is fine for testing departure from linearity. But we don’t really expect it to follow a cubic curve. Let’s fit a “modified lowess smooth” to these data, which doesn’t assume a particular form for its changes, to get this (dashed lines show 2sigma ranges):

That looks more like it slowed down around 1950, then sped up around 1990. We can sharpen our estimates of when those might have happened by fitting a continuous piecewise-linear model, and that will also test the statistical significance of those changes. The best model is this:

And yes, the changes are statistically significant. That doesn’t mean it’s following the continuous piecewise-linear model exactly — but it did decelerate early and accelerate later, and this model is at least a good approximation. It’s also very similar to the model from smoothing the data, obvious when we plot them both together (piecewise linear in blue, smooth in red):

Both models also provide an estimate of the rate of sea level rise over time (of course the piecewise linear model has a constant rate over each piece). Here they are (piecewise linear in black, smooth in red, again with 2sigma ranges above and below):

We can see that they agree overall (although the piecewise-linear model is constrained to constant rate over each piece). We can also see that the current rate of sea level rise is around 5 mm/yr, faster than the global average rate.

One of the things I’ve mentioned many times is that not only does sea level not follow a straight line, it also doesn’t show constant acceleration. It shows a much more complex pattern, and deceleration early followed by acceleration later is a feature of global reconstructions (e.g. that of Church & White), not just of individual tide gauges. For some reason, deniers just can’t seem to wrap their heads around that. They seem to insist on the simpleton’s view of sea level rise — that if it isn’t showing constant acceleration through all time, then global warming is just a fraud and we have nothing to worry about.

Such a pattern also agrees with empirical models of past sea level change, particularly those of Vermeer and Rahmstorf. The observed pattern not only shows in the data, it also matches models! For some reason, deniers just don’t want to acknowledge that.

As for Boston, one of the reasons it shows such a fast rate of increase may be the slowdown of the AMOC (Atlantic Meridional Overturning Circulation, sometimes called the “Gulf stream”). What the AMOC does in the next few decades can affect the U.S. east coast profoundly. It’s unlikely that AMOC will “shut down,” but if it does that will be incredible trouble for a lot of places. Let’s hope it doesn’t happen.

As for J. Richard Wakefield’s comment saying that “Sat data shows sea level rise doubled in rate in 1990, 100% acceleration in one year. And not one surface station shows this? Explain.” Perhaps Mr. Wakefield will explain how the satellite data show sudden acceleration in 1990 when they don’t even begin until 1993.

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31 responses to “Boston Sea Level

  1. Really clear analysis. Just one gauge but tells the story repeated gauge after gauge (with the few exceptions due to mainly vertical land movement effects). In that note it’s worth noting that even gauges far from present ice melting can see important vertical land movements that change with time. See this recent paper

    Much smaller than the signal but important caveat that gauges don’t just measure the ocean.

  2. Good to have you back, Tamino. Thanks yet again for another clear analysis.

    The denialati want it simple. In some cases, perhaps, because they know perfectly well about the effects of the Einsteinian ‘too simple.’

  3. OK so assuming this is from sea level rise and not a local caused short term phenomenon, do a few other stations, like in Australia and Brest, France. Both very long measurements.

    Now, as for this specific case, sat data shows the rate double of station data. Why is this doubling not showing in your graph? What is the rate of acceleration according to your calculations?

    [Response: Your first comment included the definitive statement “shows NO acceleration” — and everybody could tell that you arrived with a chip on your shoulder and intended to be combative. Now that you’ve been proved wrong, you resort to “do a few other stations.”

    Thanks for the wonderful opportunity to do so. But first, I’ll show the tide-gauge based data after 1993, as I said I would.

    You might look again at the plot of estimated rates at Boston. Notice that after around 1990, it more than doubled.

    One wonders whether or not you know what the satellite data say about sea level rise *at Boston* (rather than globally). I do.

    As for “a few other stations,” don’t panic. I’ve got this. But it would be polite if you would at least wait before moving the goalposts again.]

    • I guess Tamino may cover this, but the data for the Church & White global tide gauge sea level reconstruction you are referencing is here.

      Trend from 1930-1992 is 1.76mm/yr (a 0.78 multiplier is applied to the figure you reference to get 1.4mm/yr). The trend from 1993-2013 in the same tide gauge dataset is 3.56mm/yr. Double the 1930-1992 trend.

    • J. Richard Wakefield

      You have proved nothing. You’re doing your best to grasp at the acceleration straw by forcing acceleration trends on data that shows no acceleration. This is your OPINION of the the data. You have shown no facts at all.

      [Response: When I apply rigorous statistical tests, you call it my “opinion” of the data. When you sate “NO acceleration” as though it were an absolute fact, you provide nothing but a link to a graph.

      You match Neil deGrasse Tyson’s definition of a “denier”: a skeptic questions claims, then embraces the evidence. A denier questions claims, then rejects the evidence. Thanks for the clear illustration.]

    • J. Richard Wakefield

      Here is another plot of the sat data. I didnt make this. This comes from your side.

      National Geographic agrees with me.
      “However, the annual rate of rise over the past 20 years has been 0.13 inches (3.2 millimeters) a year, roughly twice the average speed of the preceding 80 years.

      NASA agrees with me, flat 3.4 since 1992:

      These sci refs agree, doubled in 1990-1993:

      “The global trend since 1995 exceeds 3 mm yr−1 which is consistent with altimeter measurements,”

      “When the entire record is assembled, the average rate of sea level rise from 1993–2009 is 3.4 ± 0.4 mm/year.”

      “Since the early 1990s, sea level rose at a mean rate of ~3.1 mm yr−1”

      This paper drops the rate from the 3.4mm/yr, however, it is a jump in the rate since 1993.
      “First, the GMSL rate (1993 to mid-2014) is systematically reduced to between +2.6 ± 0.4 mm yr−1 and +2.9 ± 0.4 mm yr−1, ”

      Seems to me you agree, as I do, that it is impossible to instantly double in just a few years and then be flat. The sat data disagrees with station data by double as the station data does not show any kind of inflection point in 1990-1993.

      And now we know why sat data was wrong. It wasnt calibrated properly.

      To get to a 2 meter rise by 2100 would require an acceleration of 3.7% per year (compound growth, the graphs the IPCC uses for their projections) . Because acceleration has a doubling period, with each doubling period the sum of all previous periods combined, then that means the rate of sea level rise in the year 2199 would have to be 80mm per year. Impossible.

      [Response: National Geographic (your first choice) says “roughly twice the average speed of the preceding 80 years.” It doesn’t say “twice the speed at 1993.” You don’t seem to get that.

      You also fail to understand what acceleration means. It doesn’t mean that acceleration has to be constant. It’s beyond the simpleton’s view.

      Not to mention, your disbelief that it can have doubled in a single year is nothing more than “argument from incredulity.”

      Considering that the rate 1993-2013 from tide gauge records is about the same as from satellites, do you now disbelieve the tide gauge records too? I see … if you don’t like it, you don’t believe it.]

  4. Nice.

    @J Richard Wakefield should note that there are stations, e.g., in Alaska, which show sea level decrease, and this is just as strong an indicator of ice sheet melting and climate disruption as rise elsewhere, due to gravitational effects.

    The near-Boston AMOC slowdown is certainly complicated. There’s the cold blob southeast of Greenland, and its genesis is still being explored. One hypothesis is cold, fresh and so light water from the Arctic, but there could be other reasons. But it is in the way, and so, there’s an interpretation that the AMOC is decelerating and, as any fluid which does that, it expands in all directions, including up.

    But this is too simple, I think, a billiard ball table explanation, even if it ends up being predominantly true. One of the wildcards in dynamics of oceanic flow are eddies, and some of these are big. It’s possible — very much open to study still, as far as I know — that with more thermal energy in oceans and more flow, eddies are tighter, capturing more energy, and there could be a kind of self-blocking stall going on. It’s hard to talk about this qualitatively. Or, for that matter, in terms of linear projections.

    One of the beauties of the presentation Tamino shows above is how to systematically simplify a complicated thing. Here’s my dynamic linear model take on Church & White.

    • J. Richard Wakefield

      Alaska’s drop in sea level is because that is an active subduction zone, the land is rising because the Pacific Plate is pushing up the land. Not because of isostatic rebound.

      [Response: Wow. Just “wow.” Are there no limits to how much you will deceive yourself? Apparently not.]

      • J. Richard Wakefield

        What is wow is your denial of geology.

        [Response: Nobody said Alaska wasn’t near geologically active zone. But that’s not the reason the land is rising so fast. It’s isostatic rebound. Or perhaps you think isostatic rebound doesn’t exist?

        The now-famous “British Doctor’s Study” established the link between cigarette smoking and lung cancer. Will you now say that the people it found who got lung cancer, got it because they were British doctors? If I respond “wow … just wow” will you retort with documentary evidence that they were British doctors?]

      • @J Richard Wakefield,

        Two words: Time constants. How rapidly do you think either isostatic rebound or plate motions operate? Rebound operates on 10,000 year time scales, and the main effects result from momentum left over from the Wisconsin deglaciation. Accordingly recent ice mass changes will not influence it much year-over-year. There are significant components due to GIS, but these varying strongly in a region due to topography. There are also gravitational variations due to mass loss from icesheet melt, e.g., near Greenland. Plates in western North America move 1-10 cm per year, laterally. Vertical components due to tectonic motions in Alaska are at most 1 mm per annum (see section 3.2 of link). Nevertheless, Skag, Alaska sees sea level dropping nearly 6 mm per annum.

        There’s a good NPS paper which illustrates the effects of GIS in Alaska.

      • J. Richard Wakefield


        [Response: Before you comment again, read the latest post. I insist.]

      • J. Richard Wakefield


        During the last glacial period, Alaska didnt have much ice on it at all. Hence cannot be rebounding as much as, for example, Hudson’s bay.

        [Response: Keep digging.]

      • @J Richard Wakefield,

        From here:

        Apparently, Mr Wakefield, you did not have the courtesy of checking a reference I provided.

      • J. Richard Wakefield

        “established the link between cigarette smoking and lung cancer.”

        I love it when you alarmists bring up cigs and cancer. How was that established? Over years and thousands of double blind lab experiments. Can you point to any double blind lab experiments on sea level rise? Nope. Not possible to do. So please, stop with these bogus comparisons, it’s not showing you much favoritism,

        [Response: Odd that you don’t address the issue: that your claim is specious. The analogy is with your behavior. And it’s “apt.”]

      • @J. Richard Wakefield

        I love it when you alarmists bring up cigs and cancer. … etc.

        Yet another example of a phenomenon that I have frequently observed: “Skeptics” never, ever understand analogies (or they pretend not to). They laser in on what’s different and completely ignore what’s the same.

        It’s an interesting pathology that should be studied.

      • Mr. Wakefield: Re. “Over years and thousands of double blind lab experiments.”

        Please provide even a single reference to a double blind, random assignment experiment which shows that smoking causes cancer in actual living humans. I will donate $100 to a charity of your choice if you can show even one which fully meets the criterion of being an example of such an experimental design.

        You seem unaware that tobacco sscience denialists used the fact that such an experiment was not possible to continually argue there was no experimental proof that tobacco causes cancer, only possibly–even probably–spurious correlations that proved nothing.

        We’ll let other stats/experimental design experts here be the judge.

  5. J. Richard Wakefield

    What on earth are you talking about? Where does your statement “sat data shows the rate double of station data” come from? For a global comparison look at the news item at the top of: and also the graph further down the page.

    If you are going to compare rates it is important to compare rates over the same period!

    Neil White

  6. Greg Simpson

    You really led down the garden path with this one. In the middle of your article I’m thinking “I can’t see a quadratic deviation from a linear trend. Maybe you can demonstrate it with analysis, but it can’t be more than just barely significant. It look more like a cubic.” And as I keep reading you come out and say it’s cubic.

    Very engaging writing.

  7. As for J. Richard Wakefield’s comment saying that “Sat data shows sea level rise doubled in rate in 1990, 100% acceleration in one year. And not one surface station shows this? Explain.” Perhaps Mr. Wakefield will explain how the satellite data show sudden acceleration in 1990 when they don’t even begin until 1993.

    Intellectual honesty should have compelled an immediate and clear acknowledgement of the mistake. Some people seem to have this idea that admitting error weakens their position. For reasonable people, it establishes trust that it’s worth continuing the discussion.

    First opportunity missed. I believe in second chances.

    • J. Richard Wakefield

      Sat data is here: Straight line starting in 1993, of a doubled rate compared to station data.

      [Response: You should read the latest post. The average rate since 1993 is not double that compared to station data.]

      • J. Richard Wakefield

        I posted science refs that disagree with you, and state categorically sat data is doubled station data. Sat data had to be readjusted because of the disagreement.

        [Response: Satellite data agree about the trend rate with the tide-gauge based data, over the period in question. For you to claim “Sat data had to be readjusted because of the disagreement” is both nonsense, and probably a set-up for you to attack the data itself.]

      • Sat data is here: Straight line starting in 1993

        You’ve now missed several opportunities to openly concede your error (“starting in 1990”) and blithely updated without acknowledging what was pointed out to you. So much for intellectual honesty.

        You say, “The sat data disagrees with station data by double.” But in the same post you link to this and quote:

        The global trend since 1995 exceeds 3 mm yr−1 which is consistent with altimeter measurements

        Did you even understand it? The trend referred to here is the tide gauge trend. Consistent with the satellite record.

        BTW, that was from a paper (thanks for the link), that was submitted for publication in July 2011. The sat data trend at that time was about 3mm/yr. Here’s an archived web page of the satellite trend as of May 2011.


        Double? No, virtually the same.

        You’ve linked a few graphs based on the work of Church and White. But you have not read the sources. What do they conclude?

        After correcting for the GIA, the linear trend from the altimeter data from January 1993 to December 2009 is 3.2 ± 0.4 mm year−1… The reconstructed global average sea-level change over the same period is almost the same as for the altimeter data. However, as a result of different interannual variability, the trend of 2.8 ± 0.8 mm year−1 is smaller but not significantly different to the altimeter estimate after correction for glacial isostatic adjustment.

        There’s a graphic of the comparison, too.

        On what, then, are you basing your opinion that “The sat data disagrees with station data by double”?

  8. J. Richard Wakefield

    [Response: Satellite data agree about the trend rate with the tide-gauge based data, over the period in question. For you to claim “Sat data had to be readjusted because of the disagreement” is both nonsense, and probably a set-up for you to attack the data itself.]

    Guess you havnt been keeping up.

    “Here we show that the rise, from the sum of all observed contributions to GMSL, increases from 2.2 ± 0.3 mm yr−1 in 1993 to 3.3 ± 0.3 mm yr−1 in 2014.”

    They adjusted the early TOPEX data down as being too high (3.4 in 1993), because, these authors claim, it wasnt calibrated properly. So instead of a 100% increase in 1993, these guys adjusted the data such that it’s just a 50% increase in 1993. So all those linear sat graphs are wrong.

    So now they are artificially manipulating data to give the desired results the dogma demands is supposed to be happening.

    [Response: The revision brings it into better agreement with the surface record. But as I have shown (you will never believe), their estimated trends since 1993 were not in disagreement.

    Odd that you haven’t admitted, or considered the implication of, your statement the the satellite data show a rate of 3.4 mm/yr over the period 1993-1995, when it actually shows 0.4. One of the denier principles: never admit your mistakes, just move and and don’t talk about it.

    But it’s no surprise you’re moving into the “accuse them of artificially manipulating data.” The 2nd-to-last refuge of scoundrels?

    Before you comment again, read the latest post. I insist.]

    • Re. “manipulation”?

      If you were a scientist–fat chance, but just play along–and 2 methods of measuring the same thing gave different results, would you 1) simply do nothing or 2) look for reasons in the measuring procedures oh like calibration errors in one instrument or another that could lead to the discrepancy?

      Yes, if you were unscrupulous and stupid you could 3) simply make stuff up. But in a field where there are many other workers such behavior wouldn’t last past peer review or 1 or 2 public statements of your methodology.

      Of course, as you propose, “they”–where “they” would have to include every tide scientist in the world–could 4) engage in a conspiracy to hoax the world for some strange political reason that applies globally. But I think your evidence for that is about nil.

    • Mal Adapted


      Of course, as you propose, “they”–where “they” would have to include every tide scientist in the world–could 4) engage in a conspiracy to hoax the world for some strange political reason that applies globally.

      They would have had to launch their hoax before 1920 and keep the secret to the present day. Scientists need to be disciplined, but they’re not so disciplined that three may keep a secret for 97 years if less than two are dead!

  9. Susan Anderson

    I live a few blocks from Tamino’s station. Sadly, I didn’t get up and get photos last night when we had a high tide combined with northeaster conditions, but over the last four years I have a variety of photos of overflow. This is interesting to me because it’s practically in my dooryard. I’ve been here since the early 1980s, and I don’t need numbers to show me that Tamino is exactly right. His science’s agreement with my direct knowledge makes him intelligent and honest in my book.

    Nobody arguing from the point of view that they need an absolute correspondence of data points with some mathematical formula or they won’t believe the water is rising is thinking straight. Science observes and seeks to understand trends and does a dam’ good job. Words and numbers are in the service of understanding, not the reverse. Nature is erratic, but vast reams of evidence from multiple types of observations are combining to confirm the best expert scientific opinion.

    People – real people – are in the target zone of all this stuff, not just on the coasts but inland. And extremes are increasing all too rapidly. When unskeptical “skeptic” deniers begin to acknowledge what is happening, it will already have gone much further than it should have in a community of intelligent humans working to solve problems.

    • Mal Adapted

      Outstanding comment, Ms. Anderson, thank you! You’ve put your finger on a fundamental requirement for not fooling oneself, namely humility before Nature. AGW-denial is refuted thus!

      I said on RC the other day that hubris, that is, rejection of the mediocrity principle, is presumably what underlies some trained scientists’ “dissent” from their peers’ consensus for AGW.

      IM assuredly HO, some scientifically credentialed AGW-deniers implicitly, if not explicitly, fool themselves that economic growth fueled by fossil carbon is divine providence, magically bestowed on an entitled humanity without a hidden price tag. It’s not scurrilous to allege that they cherish the illusion of their own and their species’ privileged status in the Universe, are proud of their contributions to human material prosperity, and are offended that anyone would challenge their entitlement or complain about their unpaid bills.

      It’s not surprising that non-scientifically-qualified AGW-deniers, many of whom cherish the same illusion, are receptive to the tiny minority of published climate scientists that deny AGW to one extent or another.

      • Susan Anderson

        Thanks Mal. You’re not so dusty yourself. Keep thinking, and keep your chin up!