Are We Chasing Imaginary Numbers?

Guest Essay by Kip Hansen

 Spoiler Alert: This essay is not about the mathematical entity the imaginary number. I do think that an essay here about imaginary numbers of that sort would be interesting, but this isn’t going to be it. This essay, while not about the usual fare seen here – AGW; CAGW; Catastrophic Climate Change; Global Cooling; various oxides of carbon; the pH, level, or surface temperature of oceans; or the antics or ethics (or lack of ethics) of various international scientists and politicians — will hopefully be interesting to the majority of readers. It will ask more questions than it answers.

Last Saturday, 3 October 2015, WUWT’s indefatigable Willis Eschenbach published a guest essay regarding an NPR radio report by Ira Flatow that labelled “some recent pictures of flooding in Miami, Florida, as evidence that climate change is real and is already affecting Florida.” In response to a comment I made to that essay, Willis asked this very interesting question:

“…as you say, we can measure sea level with a “high degree of accuracy” … so are we measuring an imaginary thing? And if we average those highly accurate measurements, why would we not get a global average sea level? What am I missing here?”

In science, asking the Right Question is often, maybe always, more important than having the Right Answer. Let’s look at Willis’s questions and see what we can find out about the world and the world of science.

What are the questions here?

1. Can we measure sea level with “a high degree of accuracy”?
2. Are we measuring an imaginary thing (when we do so)?
3. If we average those highly accurate measurements, why would we not get a global average sea level?
4. What am I [we] missing here?

It is my idea here to ask a more generalized question — what are we measuring in Climate Science and are we measuring an imaginary thing when we do so? — but we can use “sea level” as the thought experiment example.

Let me address the first question first: Is it really possible to measure something like sea level (or surface air temperature 2 meters above the ground or sea surface temperature) with “a high degree of accuracy”?

When I stated in my original comment that we had been measuring sea level with a high degree of accuracy for years, I meant that we knew what sea level could be expected at various places at future times and had an idea what a more generalized “global sea level” might be and what changes had been seen over longer time periods like the last century or so. But for our thought experiment in this essay, let’s define “high degree of accuracy” as the commonly mentioned “annual anomaly” in the scientific literature. For “global average sea level” this is in single digit millimeters, usually 1.7/1.8 up to 3.4 mm per year, somewhere in that range. (For those thinking along on other paths, that might be tenths and hundredths of a degree Centigrade for global average surface air temperature and sea surface temperature, and even smaller, thousandths of a degree C for ocean water temperatures leading to a calculation of ocean heat content.)

Before we get very far, let’s ask “Why do we [they] want to measure global sea level?” The major reason seems to be, in our politicized world of global warming politics, that many want to measure global sea level to show that it is rising (which it has been for quite some time, at least the last 20,000 years) and that this continuing rise is 1) dangerous and 2) due to recent surface temperature rise over the last century, thus 3) due to Global Warming.   The theme is to use sea level rise as a proof of increased thermal expansion of the water in the oceans and increased addition of water from melting land ice deposits, both asserted to be the result of Global Warming caused by increased atmospheric concentrations of greenhouse gases, primarily CO_2, since the 1880s . We’ll see later in this essay that this is part of a larger modern scientific movement to produce “single numbers” to represent dynamic systems (some of which are properly known to be nonlinear dynamical systems).

Can we measure sea level to that (+/- 3 to 4 mm) degree of accuracy? Well, for sea level, even at a single precise location, the answer is “No, we can not.” Now, I am not trying to be provocative here, it is a simple matter of fact. If the sea would be so kind as to stand still, even for just a few moments, we could get in a very accurate measurement at a single spot, or even a lot of spots.   Alas, the sea is never still, it is always moving up and/or down: tides, currents, wind chop, waves, wakes of passing vessels, rising and falling air pressure and, in most important locations, all of those at once. Thus, we cannot physically do it; the sea does not stand still long enough for us to make this measurement to that degree of accuracy. This gets only worse when we add in the information that both the dry land itself and the bottoms of the oceans, almost everywhere, are also in vertical motion and busy changing the volume of the ocean basins.

Many will protest: “Look here, Mr. Hansen. You can’t say that. There are scads of very scientific tables, charts, and journal articles very carefully telling us that now only can we make that measurement, we have been doing so for much of human history and [drum roll, please] since 1992 with [gasp!] satellites!”

It is my point here that what we are doing, where the doing is done, is not measurement, but derivation. Many measurements are taken, in many and diverse locations, at many and diverse times. In some cases, there are nearly continuous time series of measurements for particular locations. From these numerous individual measurements, for example, the tide station reports from the Battery in New York City, an interesting (but not to be detailed here) formula is applied to derive a figure, a single number, that represents the average difference between the sea surface and a geodetic bench mark (set in the bedrock of Manhattan Island years ago) over some period of time. We will skip the nearly infinite details as to whether the derived number represents a simple average between highs and lows, or is an average against time.

Let me point out that the NOAA CO-OPS system of tide stations has a very important and pragmatic purpose. Ships and boats need to know the depth of the water they will find in a particular spot – at a dock on the Hudson River or over the sand bar across the inlet – and at a particular time. Thus, tide tables are very important to sea going commerce and recreational boaters. It answers important questions such as: “Can I get there without hitting those nasty rocks (or going aground on that sticky mud) on the bottom? Can I stay here without being set down by the tide on those rocks or mud?” This system was never designed to measure “sea level rise” nevertheless it is used to compute changes in relative sea level trends in ports of American interest. Here are two Wiki articles on sea level: here and here. In the second article, this image is shown:

Notice please the difference between the trend calculated from tide gauges (orange line with grey error range) and the blue satellite measurements. Tide Gauge data (which measures Relative Sea Level at each tide gauge) accelerates while satellite data, which measures absolute sea level, keeps to its century long trend.

But what of those marvelous satellites? The official NOAA claim is: ”A series of satellite missions that started with TOPEX/Poseidon (T/P) in 1992 and continued with Jason-1 (2001–2013) and Jason-2 (2008–present) estimate global mean sea level every 10 days with an uncertainty of 3–4 mm.”    Results can be seen on graphical form at NOAA’s Laboratory for Satellite Altimetry web site. It is interesting to see the difference in visual impact that results from the use of alternate coloring schemes and to observe the lumpiness of the oceans.

I know many of the readers here are familiar with the sea – Willis and I have each spent a hefty fraction of our lives living on the sea, and an ever greater fraction living at the edge of the seas. Three to four millimeters is between 0.12 and 0.16 of an inch – about the thickness of two American pennies stacked atop one another. Or, for our cousins in the United Kingdom, about as thick as a one pound coin. It is a rare and beautiful and awe inspiring sight to see the ocean smooth as glass to the horizon, or even just across the bay or harbor. In my one-third of a lifetime of living on the sea (totaling > 20 years), I have only occasionally seen the sea so smooth – the slightest breezes bring up wind ripples and chop that far exceeds 3-4 mm, and can build quickly to feet and meters. If a body of water is open to the ocean, undulating ocean swells march from one horizon to the other, swells also measured in multiple feet or meters, and not necessarily traveling in the same direction as the wind chop. This all adds up to a great deal of vertical motion of the sea’s surface – at times exhilarating and at times downright frightening.

Now if NOAA wants to claim that their satellites in their perfect orbits can somehow transmogrify the undulating, rising and falling, uneven surface of the Earth’s ocean to a resolution of +/- 3 to 4 mm, then very well. Who am I to say they can’t, even if I can’t imagine how they might even theoretically do so. Nonetheless, for our purpose here, let us make this distinction: they do not measure “global mean sea level every 10 days” – they don’t even claim to, their claim is that they estimate it. In every real pragmatic sense, they somehow derive a single number from a fabulously massive amount of data – data which in and of themselves are not direct measurements, but inferences of measurements made from other kinds of data.

Let’s quit fooling around. While it would be possible to measure sea level in individual locations, it is difficult and even when done it is not a true measurement, but a derivation from accumulated data and dependent on mathematical and statistical methods and definitions. If you ever find a particular section of sea at “sea level”, it will be totally momentary and accidental.

Sea Level, even “Sea Level at the Battery in New York”, is not properly represented by a single number, above and below some geodetic bench mark. What we call sea level is a derived, calculated number – an average of averages of an array of measurement time series. In this sense, as the calculated mid-point of a range over time, it is, in a practical sense, an imaginary number having no existence in the day-to-day life of the Port of New York.

There is, however, a pragmatic “sea level at the Battery in New York” – which itself is a predictable range above and below some depth of water at a certain point (a point referred to as Local Mean Sea Level) which, when modified by information of expected, predicted tides, can be extrapolated to other points in the harbor, which is useful for mariners despite its less-than-real aspect. It can be used in its gross form (fractions of feet or meters) to determine the depth of water over the bottom at a place and time important to a ship’s captain and crew. Here is the prediction of water levels, relative to MLLW, made for October 9^th thru October 11^th.

The bottom line is that sea level, anywhere and at any time, is not a direct measurement. Never. It is a calculated, derived number that represents a precisely defined, but actually quite complicated, idea.

In order to define global sea level, one must participate in an exercise of imagination along the lines of: Imagine that the planet has stopped spinning; that moon has never existed; that the planet is a perfect sphere (or perfectly regular ovoid or flattened sphere); that there is no wind; that the atmosphere is evenly distributed and air pressure is the same at all points; that the temperatures of the seas are all exactly even, everywhere, to all depths; that there are no currents;, that there are no ice caps; that the rivers have stopped flowing into the sea and that gravity is magically equal at all points on the Earth’s surface (it is not, btw): under those conditions, we could then say that global sea level would be precisely “there”, within 3 or 4 mm. My friends, this is what makes Global Average Sea Level, in this special sense, an imaginary number.

So, we have answered Question 2: Are we measuring an imaginary thing (when we do so)?  Yes, we are “measuring”, in a sense, an imaginary thing when we say we are measuring sea level. The resulting calculated, derived number is a creature of our imaginations, an imaginary number.

Question 3 almost answers itself. “If we average those highly accurate measurements, why would we not get a global average sea level?” One can carry out a dizzying number of statistical and mathematical steps and arrive at some number – the more division steps involved the more precise looking the number will be. One can average any set of numbers. In this case, will one arrive at a number that is the “global average sea level”? Let’s look at Question 4 first and come back to this.

Question 4 is “What am we [originally “I”] missing here?”

This is a question of logic, and kind of follows on from an earlier essay I published here in February regarding Uncertainty Ranges.   When one averages a series of numbers that are in reality themselves ranges, then the result must also be a range. In our case today, when averaging a series (or in this case, a computer-full) of imaginary numbers then the result must be another imaginary number, in the same sense as the numbers in the original data set.   You can not average away original measurement error, you can not average away the fact that data given are themselves really ranges rather than single numbers, you can not average away the fact that original numbers themselves are, in the sense discussed here today, imaginary.

Before we too far afield here, let’s try to be clear on what the distinction is between a real number and what I have been calling here an imaginary number. This discussion takes place in the context of the measurement of characteristics of the physical world. For the result of a measurement to be a real number, the thing being measured must itself be measurable and the numerical result representing that measurement must represent something that exists in some meaningful and useful sense. However, the result of a measurement of a thing that itself is not physically measurable, but which can only be derived mathematically based on a definition that itself is an object of our imaginations (not something actually found in the real world), then that result should itself be considered, in this special sense, imaginary as well, despite its seeming precision.

There are innumerable averages of things that can be derived and calculated. Despite that, many of those averages are themselves imaginary, and their meaning and usefulness must always be thoughtfully considered. Such imaginary numbers may have some interesting meaning and some pragmatic usefulness but great care must be taken with their application, because, after all, they are imaginary and do not exist in reality.

Thus the average height of American citizens can be useful in determining the sizes of beds sold to Americans, at least indicating a range to be considered, it would be foolish to declare it the proper height of doorways for all new construction, even with an inch to spare tacked on, or to make exaggerated, scary, claims about public health threats based on the tiniest changes in such a number over some narrowly-selected time period.

Worse yet, and I hope there will be some comments in support of at least this idea, simple averages of averages of averages (all of which start with averaged, imaginary, derived numbers rather than actual measurements) are abominable absurdities. [ref: Simpson’s Paradox, etc.]

Here’s a ridiculous example: If we calculated the average altitude of the land in the state of South Carolina, first averaging the altitude of each county, then averaging the altitude of multi-county regions, and finally averaging regional altitudes, the result would be a number like (a totally pulled-out-of-the-air guess) 125 feet above sea level and when trended from the highest point in the Blue Ridge Mountains to the sea the state could be said to have a slope of XX feet per mile. It makes no difference in this sense if we weight the averages, krig the missing points, homogenize or smooth or smear. This procedure calculates and/or derives an imaginary number in the special sense of our working definition here. Thus, with our magic new imaginary number, it might be claimed that while some areas of South Carolina could be flooded by extreme high tides simultaneous with two feet of rain, on average the people there would not be prone to disaster as even the few expected flooded areas would quickly drain into the Atlantic.  Applying such a totally mathematically correct yet imaginary number to the real world can result it wildly inappropriate conclusions. It was this type of logic powered by imaginary numbers that led a New York Times science journalist to erroneously claim that the global sea level rise caused by global warming (a real rise but an imaginary number) caused increased damages to New York City during Hurricane Sandy — the same error Ira Flatow made in the NPR segment about flooding in Miami, where the flooding referred to occurs at a spot that is below the long-term Mean High Tide, and was so when the street was constructed.

Now, coming back to Question 3:  “If we average those highly accurate measurements, why would we not get a global average sea level?” If we average the very large data set of imaginary numbers for a specific moment in time, we will arrive at a new, even more imaginary, single number that could be called, if everyone were willing to allow it, “global average sea level”.   Would it be pragmatically, practically, meaningful and useful? Maybe, but in a very limited sense…and we would have to be very careful as to what meaning we assigned to it.

Why? See my essay last year about Hurricane Sandy and damages to NY City. The purported sea level rise for the 50 year period 1960 – 2010 “caused by global warming driven sea level rise” should have been 4 inches (roughly half of the 8 inches over the last century). In actuality, only when we use the lowest estimate of subsidence for the Battery couple with the highest estimate of local relative sea level change do we see any positive contribution of absolute, global sea level change to the relative sea level at the Battery, the 0.59 inches in the upper right-hand corner:

What’s up here? The acknowledged century-long estimated global sea level rise did not show up at the Battery, not even over the most recent 50 year period. This should not surprise us – attempts to apply a single-number, “global sea level rise”, is ill-thought out – trying to apply an imaginary number to a specific real situation.

Today’s discussion is one way of looking at the current trend in Science in which attempts are made to reduce very complicated dynamic systems to a single number which can then be graphed against time, usually in attempts to do one or more of the following:

1. to cast blame for the increasing or decreasing number on a substance or action or group, usually incorrectly
2. using two such graphs of single numbers to correlate some single number with some other single number to sell a desired story, usually to cast blame or give credit, usually incorrectly
3. to bring attention to [read this as: to cause public concern or worry about] some rising or falling single number in hopes of generating gain [in research funds, fame, public sympathy, public or political support], usually unwarranted

These single numbers, meant to somehow illuminate some feature of the real world, are often, maybe almost always, not real numbers representing real things, but imaginary numbers representing concepts that exist, on a pragmatic practical level, only in our imaginations, which may lack meaningfulness and usefulness, or both. In this special sense, we can rightly refer to them as imaginary numbers. And because they are almost never acknowledged as imaginary numbers which require special care in application, each of the three uses above is followed by “usually incorrectly” or “usually unwarranted”.

Now, even if you don’t agree with me, it should be interesting to discuss in comments some of the ongoing efforts to [mis-] use this special breed of derived number, the imaginary number, to sway public opinion in differing scientific fields around the world. I’d really like to hear your views and benefit from your experience.

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Author’s Comment Policy: This essay is not really about global sea level, but I doubt we’ll be able to discuss it without also touching on the issues surrounding the issue of global mean sea level. I do know something about it and will try to answer questions.

I’d rather discuss the concept of “Are we chasing imaginary numbers?”

It’s just an idea…let’s talk about it.

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