Decimals of Precision

Guest Post by Willis Eschenbach

Over at Judith Curry’s excellent blog there’s a discussion of Trenberth’s missing heat. A new paper about oceanic temperatures says the heat’s not really missing, we just don’t have accurate enough information to tell where it is. The paper’s called Observed changes in top-of-the-atmosphere radiation and upper-ocean heating consistent within uncertainty.

It’s paywalled, and I was interested in one rough number, so I haven’t read it. The number that I wanted was the error estimate for their oceanic heating rates. This error can be seen in Figures 1a and 3a on the abstract page, and it is on the order of about plus or minus one watt/m2. This is consistent with other estimates of upper ocean heat content measurement errors.

I think I can conclusively demonstrate that their claimed error is way too small. To understand why, let me take a detour through the art, science, and business of blackjack.

In a fit of misguided passion, some years back I decided to learn how to count cards at blackjack. I had money and time at the same moment, an unusual combination in my life, so I took a class from a guy I’ll call Jimmy Chan. Paid good money for the class, and I got good value. I’ve always been good with figures, and I came out good at counting cards. Not as good as Jimmy, though, he was a mad keen player who had made a lot of money counting cards.

At the time they were still playing single deck in Reno. And I was young, single, and stupid. So I took twenty thousand dollars from my savings for my grubstake and went to Reno. It was an education about a curious business.

Here are the economics of the business of counting cards.

First, if you count using one of the usual systems as I did, and you are playing single deck, it gives you about a 1% edge on the house. Not much, to be sure, but it is a solid edge. And you can add to that by using a better counting system or a concurrent betting system, where better means more complex.

Second, if you play head-to-head (just you and the dealer) you can typically play about a hundred hands an hour.

Doesn’t take a math whiz to see that if you don’t blow the count, you will win about one extra hand an hour.

And therein is the catch. It means that in the card counting business, your average hourly wage is the amount of your average bet.

It’s a catch because of the other inexorable rule of counting blackjack. This regards surviving the swings and arrows of outrageous luck. If you don’t want to go home empty-handed, you need to have a grubstake that is a thousand times your average bet. Otherwise, you could go bust just from the natural ups and downs.

Now, twenty thousand dollars was all I could scrape together then. So that meant my average bet couldn’t be more than twenty dollars. I started out at the five dollar level.

I’d never spent any time in a casino up until then. I felt like the rube in every movie I ever saw. I played a while at the five dollar level. You never win or lose much there, so nobody paid any attention to me.

After a day or so making the princely sum of $5 per hour, I started betting larger. First at the ten-dollar level. Then at the twenty-dollar level. That was good money back in those days.

But when you start to make a bit of money, like say you hit a few blackjacks in a row and you’re doubling down, they start paying attention to you, and the trouble begins. First they use the casino holodeck to transport a somewhat malignant looking dwarf armed with a pad and a pencil to your table. He materializes at the shoulder of the dealer, and she starts to sweat. I say she because most dealers were women then and now. She starts to sweat because the casino doesn’t really care about card counters. I was making $20 an hour on average? Big deal, everyone in the casino management made that and more.

What scares casino owners is collusion between dealers and players. With the connivance of the dealer a guy can have a “string of luck” that can clean out a table in fifteen minutes and be out the door, meeting the dealer later to split the money. That’s what casino owners worry about, and that’s why the dealer started sweating, she knew she was being watched too. The dwarf peered through coke-bottle thick glasses, and wrote down the number of chips on each stack in the dealer’s rack, how much money I had, how much other players had. He gave the dealer a new deck. He wore a suit that cost as much as my grubstake. His wingtip shoes were shined to a rich luster. He looked at me as though I were a rich man with a loathsome disease. He watched my eyes, my hands. I started sweating like the dealer.

If I continued to win, the holodeck went into action again. This time what materialized were two large, vaguely anthropoid looking gentlemen, whose suits were specially tailored to conceal a bulge under the off-hand shoulder. They simply appeared, one at each shoulder of the aforementioned vertically challenged gentleman, who looked even dwarfier next to them, but clearly at ease in his natural element. They all three stared at me, and when that bored them, at the dealer. And then at me again.

And if the dealer was sweating, I was melting. I’m not made for that kind of game, I’m not good at that kind of pretence. I found out you can take the cowboy out of the country, but you can’t make him go mano-a-mano with the casinos for twenty bucks an hour.

I lasted a week. I logged my hours and my winnings. During that time, I worked well over forty hours. I only made enough money to pay for the flight and the hotel, and that’s about it. I was glad to put my twenty grand back in the bank.

I couldn’t take the constant strain and pressure of counting and not looking like I was counting and trying to stay invisible and feeling like a million eyes in the sky were watching my every eyeblink and having an inescapable feeling of being that guy in the movies who’s about to be squashed like a bug. But for those who can make it a game and keep it up, what an adventure! I’m glad I did it, wouldn’t do it again.

The part I liked the least, curiously, was something else entirely. It was that my every move was fixed. For every conceivable combination of my cards, the dealer’s card, and the count, there is one and only one right move. Not two. Not “player’s choice”. One move. I definitely didn’t like the feeling that I could be replaced by a vaguely humanoid 100% Turing-tested robot with a poor sense of dress and a really, really simple set of blackjack instructions

But I was still interested in the math of it all. And I had my trusty Macintosh 512. And Jimmy Chan had an idea about how to improve the odds by changing his counting method. And so did some of Jimmy’s friends. And he had a guy who tested their new counting method for them, at some university, for five hundred bucks a run.

So I told Jimmy I’d do the analysis for a hundred bucks a run. He and his friends were interested. I wrote a program for my Mac to play blackjack against itself. I wrote it in Basic, because that was what was easy. But it was sloooow. So I taught myself to program in C, and I rewrote the entire program in C. It was still too slow, so I translated the critical sections into assembly language. Finally, it was fast enough. I would set up a run during the day, programming in the details of however the person wanted to do the count. Then I’d start it when I went to bed, and in the morning the run would be done and I’d have made a hundred bucks. I figured that I’d finally achieved what my computer was really for, which was to make me money while I slept.

The computer had to be fast because of the issue that is at the heart of this post. This is, how many hands of blackjack did the computer have to play against itself to find out if the new system beat the old system?

The answer turns out to be a hundred times more hands per decimal. In practice, this means at least a million hands, and many more is better.

What we are looking at is the error of the average. If I measure something many times, I can average my answers. Is the resulting mean value the true underlying mean of what I am measuring? No, of course not. If we flip a hundred coins, usually it won’t be exactly fifty/fifty.

But it will be close to the true average of the data. How close? Well, the measure of how close it is expected to be to the true underlying average is what is called the “standard error of the mean”. It is calculated as the standard deviation of the data divided by the square root of the number of observations.

It is the last fact that concerns us. It means that if we double the number of observations, we don’t cut the error in half, but only to 0.7 of the original value. One consequence of this is that if we need one more decimal of precision, we need a hundred times the number of observations. That is what I meant by a hundred times per decimal. If our precision is plus or minus a tenth (± 0.1) and we want to know the answer to one more decimal, plus or minus one hundredth (± 0.01), we need one hundred times the data to get that precision.

That is the end of the detour, now let me return to my investigation of their error estimate for the ocean heating rate for the top 1800 metres of the ocean. If you recall, or even if you don’t, that was 1 watt per square metre (W/m2).

Now, that is calculated from temperature readings from Argo floats, about 3,000 of them during the study period.

Let me run through the numbers to convert their error (in w/m2) into a temperature change (in °C/year). I’ve comma-separated them for easy import into a spreadsheet if you wish.

We start with the forcing error and the depth heated as our inputs, and one constant, the energy to heat seawater one degree:

Energy to heat seawater:, 4.00E+06, joules/tonne/°C

Forcing error: plus or minus, 1, watts/m2

Depth heated:, 1800, metres

Then we calculate

Seawater weight:, 1860, tonnes

for a density of about 1.03333.

We multiply watts by seconds per year to give

Joules from forcing:, 3.16E+07, joules/yr

Finally, Joules available / (Tonnes of water times energy to heat a tonne by 1°C) gives us

Temperature error: plus or minus, 0.004, degrees/yr

So, assuming there are no problems with my math, they are claiming that they can measure the temperature rise of the top mile of the global ocean to within 0.004°C per year. That seems way too small an error to me. But is it too small? If we have lots and lots of observations, surely we can get the error down to that small?

Here’s the problem with their claim that the error is that small. I’ve raised this question at Judith’s and elsewhere, and gotten no answer. So I am posing the question again, in the hope that someone can unravel the puzzle.

We know that to get a smaller error by one decimal, we need a hundred times more observations per decimal point. But the same is true in reverse. If we need less precision, we don’t need as many observations. If we need one less decimal point, we can do it with one-hundredth of the observations.

Currently, they claim an error of ± 0.004°C (four thousandths of a degree) for the annual average upper ocean temperature from the observations of the three thousand or so Argo buoys.

But that means that if we are satisfied with an error of ± 0.04°C (four hundredths of a degree), we could do it with a hundredth of the number of observations, or about 30 Argo buoys. And it also indicates that 3 Argo buoys could measure that same huge volume, the entire global ocean from pole to pole, to within a tenth of a degree.

And that is the problem I see. There’s no possible way that thirty buoys could measure the top mile of the whole ocean to that kind of accuracy, four hundredths of a degree C. The ocean is far too large and varied for thirty Argo floats to do that.

What am I missing here? Have I made some major math mistake? Their claimed error seems to be way out of line for the number of observations. I’ve not been able to find a good explanation of how they come up with these claims of extreme precision, but however they’re doing it, my math doesn’t support it.

And that’s the puzzle. Comments welcome.

Regards to everyone,

w.