It turns out that this is my 300th post, which is rather disturbing. I really should find something better to do (enough cheering in the back there). Yesterday’s post was about Ocean heat content uncertainties and it reminded that there was something I wanted to look at and maybe write about. I had some time last night (yes, my life can be boring) and so managed to do what I’d been considering.
For the last decade or so, one way of determining the ocean heat content has been to use ARGO buoys/floats. There are 3000 of these around the world’s oceans, and they drop down to 2000m and (amongst other things) measure the temperature. The temperature measurements can then be converted into an energy and these can then be combined to determine how the energy in the ocean – in different layers – is changing with time. In the last decade, it’s increased by about 1023J which means that the average temperature in the ocean has increased by about 0.05oC. The uncertainty in each temperature measurement is also about 0.05oC, which leads some to argue that ARGO buoys should not be able to measure such small changes [Correction : Mike McClory, in the comments, points out that the ARGO measurements are likely accurate to +-0.002oC, not 0.05oC, but that doesn’t really change what I’m trying to illustrate here].
Of course, this may be true when it comes to individual measurements, but when you combine lots of measurements you can measure such a small change. That’s what I thought I would try to illustrate here. Just to be clear, what I’m doing here is very simple and I’m not suggesting that it somehow represents ARGO data analysis. All I want to try and illustrate (hopefully) is that even if each measurement could not detect such a small change, summing over many measurements allows one to determine the effect of such a small change.
Let’s consider the following. Imagine we have some medium (the ocean, for example) and we have 3000 measuring devices evenly distributed throughout this medium. Let’s imagine these devices can measure the change in some quantity (energy for example) and that they can do so to an accuracy (1σ) of 1 (units don’t matter for this illustration). Let’s also consider that, initially, there is no change in this quantity (i.e., it is zero). I wrote a little computer code that would randomly generate 3000 numbers with a mean of 0 and a standard deviation (which is the 1σ error) of 1. This is shown in the figure below.
The figure above is my 3000 initial measurements. To determine the total change in this quantity (over the whole medium) I need to add all these measurements together. Normally one would then need to do some error analysis, but because I’m using a simple computer code I can simply repeat the entire experiment as many times as I like. So, I’ve done this 10000 times – I recalculate the 3000 measurements and sum them 10000 times. The distribution of the results are below. As expected, the mean is zero (i.e., the quantity is unchanged) but there is a range of values. The standard deviation of a distribution is the the distance from the mean such that 67% of the values lie within this region. By eye, it appears to be about 50. Given that my measurement errors are uncorrelated, I would expect the error in the sum to be the square root of the sum of the squares of the individual errors. These are 1, so the sum of the squares is 3000, and the square root of that is 54. So, looks pretty good and quite nicely illustrates how basic error analysis works.
At this stage I still haven’t illustrated how one can use many measurements to determine what is essentially a small change in some quantity. Let’s change the above slightly. Let’s consider a situation in which the quantity I’m measuring has increased (in each measurement cell) by an amount 0.1 (i.e., much smaller than the measurement error of 1). I can then produce 3000 measurements with a standard deviation of 1, but with a mean of 0.1. That’s shown in the figure below. It looks very similar to the first figure I showed and you’d be hard pressed to argue (by eye alone at least) that the mean is 0.1 and not 0.
Now, I redo the second part of my illustration by doing 10000 realisations of the sum of the measurements (i.e., I produce 3000 random numbers with a mean of 0.1 and a standard deviation of 1 and sum them 10000 times). The result is shown below. Now what I have – as expected – is a distribution with a mean of 300 (0.1 x 3000) and a standard deviation (1σ error) of about 50. So, even though an individual measurement would not have indicated that this quantity had increased, by summing many measurements I’ve been able to show, quite clearly, that – over the entire volume – this quantity has increased by 300 +- 50.
So, unless I’m made some kind of silly mistake, I think this illustrates how you can use many measurements to determine the change in something despite the change in each measurement potentially being smaller than the measurement error. To be clear, I’m not suggesting that this correctly represents ARGO data analysis or is even a particularly good illustration of ARGO data analysis. I’m simply trying to illustrate that arguing that you can’t measure an average increase in temperature of 0.05oC across the entire ocean because the uncertainty in each measurement is 0.05oC is wrong.
Rachel’s comment made me consider what you’d get if you took an average of the measurements, rather than the sum. So, for the case where the quantity has increased by 0.1 in each measurement volume, I repeated the calculation, but this time averaging the measurements, rather than summing, and repeating the “experiment” 10000 times. The result is below. It’s clear that the quantity has increased by 0.1 with a 1σ error of about 0.02 (which is, I think, the square root of the sum of the squares of the original measurement errors divided by the number of measurements, – i.e., because we’re averaging, each measurement can be regarded as being divided by the number of measurements and each measurement error is now 1/3000, so the sum of the squares is 0.000333 and the square root of that is 0.018). So, whether you sum or average, you can still detect a change in a quantity that, for each measurement, is smaller than the uncertainty in each measurement.