This falls in the category of odd things found while trying to answer a few questions I was asking myself. One of the questions I’ve always wondered about is: “Do the measurement errors in the global mean surface temperature look like “white noise’?” That is, to say, is the correlation between a mis-measurement in GMST in July and a mismeasurement August close to zero?
I’ve always assumed the errors would be more or less “white”. Or, failing that, the correlation would die off fairly quickly, that is, within months rather than years.
As long as that was a working assumption in my mind, I decided I’d subtract Hadley measurements from GISS measurements and create quick and dirty correlogram.
Mind you, the difference between temperature reported by Hadley and GISS don’t fully describe measurement uncertainty. After all, the two sources do draw from an overlapping set of local measurements. But, the difference does provide a sort of in situ estimate of “measurement uncertainty”. So, it figured it’s worth taking a look at some of the properties of the difference in reported values.
The first plot I made (and don’t illustrate here) showed lots of autocorrelation that decreased slowly, and also included small amplitude “wiggles” due to the annual cycle. The “wiggles” didn’t surprise me, as any slight errors in determining the anomaly for each month could result in this sort of thing. So I just found the average for each month during the period I was looking at and subtracted those. Then, I plotted the correlogram again. Voila!
The green shows the difference Hadley and GISS’s reported temperatures over land. It behaves somewhat as I’d expected. Sure– I’d expected lag1 correlation to be less than about 0.1 which is close to the 95% confidence interval for distinguishing between a value for the correlation and zero for this batch of data. But, since some of the difference between the reported temperatures is due to lack of station coverage, and some algorithmic, I thought we might see some correlation at a 1 month lag. I’d expected the correlation to drop to zero and just oscillate around zero within a few months. How few? Well… maybe two or three months? 🙂
Still, that was just a hunch. Evidently, it takes longer.
The yellow symbols represent differences in temperature averaged over the full surface of the earth (or as much of the earth as covered by each group.) The lag1 autocorrelation exceeds 0.4, and drops slowly crossing zero near 50 months. It falls to nearly -0.2 at a lag of almost 200 months, and rises back toward zero.
This surprised me.
Of course, that might just be me. Since GISS extrapolates over the poles and Hadley does not, the long period during which the autocorrelation remains positive, and the negative loop may be related to some persistent local weather patterns. (Or it might not! Maybe someone even knows what this means.)
But I hadn’t really set out to look at this as the “main thing”
In case you were wondering, the main thing I was trying figure out was the standard deviation for the difference in reported measurements over the time period in question (i.e. 1914-1942). I wanted to do something quick and dirty so I could know if the difference in data sources fell in the “large”, “medium” or “small” range relative to the likely magnitude of “weather noise”. Knowing this will eventually matter when comparing the “weather noise” in models to estimates of weather noise for the earth. (Basically, if the “noise” in measurements is nearly all due to measurements, it means there is very little weather noise. But, if the “noise” due to measurements is tiny compared to real weather noise, I can just note measurement noise is small and ignore it when comparing weather noise in models to measurements.)
What did I discover as “the main thing”? Well, if I rebaseline NOAA, Hadley and GISS to their averages from 1914-1942, the standard deviations for land-ocean merges from the three groups is about 0.04C. The standard deviation for land only measurements is about 0.11C. These differences are large enough to matter when comparing model “weather noise” to estimates of “earth weather noise” based on observations, though how much will depend on individual comparisons. So, the issue of measurement uncertainty will be discussed again.
Notes:
For these plots, I used the definition of “correlation” defined at this NIST page. An alternate definition is discussed here.
But both sets do share a lot of the same data, don’t they? They are not “independent measurements”, so you do expect quite a bit of autocorrelation. I notice that the initial drop is quite fast (the very narrow peak), and that part may be the part due to instrumental error, superimposed on the rest. I’d have to do the maths to figure that out… I know that I’ve seen that shape before, just trying to remember where, something to do with autocorrelation of short laser pulses…
It seems to me that if you were to substract CRUT and satellite, then you would have truly independent measurements, and maybe a more significant result.
Francoiso–
The sharing of data can’t cause a lagged correlation in the difference of the two measurements.
If Hadcrut is H=X+Y and GISS is G=X+Z Then
G-H= Y-Z. So, the shared measurements don’t affect that quantity. (I didn’t do the cross-correlation of <HG>. That would include the shared components.)
The thing is, G-H also isn’t precisely a measure of the uncertainty because “X” , the shared part, contains it’s own uncertainty!
Since there is no possibility of comparing to “known truth”– as required for a real calibration, we can’t really know the properties of the measurement error. (Hadley does provide some estimate, and documents their method of estimating them in a journal article.)
The satellites weren’t available before 1979.
The difficulty with comparing the surface to the satellites is that quantity could actually also mean something physical. Still, it’s worth doing, as the possibility the comparison reveals something is, itself interesting!
GISS primarily uses the “unadjusted” GHCN dataset for land measurement (http://data.giss.nasa.gov/gistemp/sources/gistemp.html) while HadCrut uses “The land-surface component of HadCRUT is derived from a collection of homogenised, quality-controlled, monthly-average temperatures for 4349 stations.” (http://hadobs.metoffice.com/crutem3/HadCRUT3_accepted.pdf)
Lucia,
I guess what I mean is that the measurement error is included in X, if both data sets start with the same measurement data, which is how I understand the difference between the two (I don’t know the detail, though). So they cancel out when you substract the two. Whatever Y and Z are (different adjustments or interpolations) is not the measurement error. So the autocorrelation properties of Y-Z are not the same as those of the measurement errors themselves. Whereas if you take satellite versus HadCrut or GISS, then you have totally independent measurement errors, which won’t cancel out. I know it doesn’t cover the period you’re interested in, but it would be worth a try just to see what it looks like (sorry, I’m too busy today to do it myself!).
Francois– I agree the difference in Y and Z isn’t purely the errors– but measurement error contributes to it.
If we define measurement error as being the difference between the thing you want to measure and the measurement you report, uncertainties in the entire process– including interpolating and adjusting, are causes of “measurement error”. The just aren’t instrumentat error.
Maybe there is a different word for that final error, but it’s the difference between the “measurement” and “quantity you measured”.
I will be looking at the difference between the satellites and surface measurements. Their errors for satellite and surface measurements are uncorrelated– but the difference is complicated by the fact that they also don’t measure precisely the same thing.
It will be interesting to look take a lok.
Lucia
the normal course of events when I read one of your posts is I quickly realise I haven’t the foggiest idea wht you’re batting on about, and resort to smugly noting all your spelling and grammar mistakes.
So it was a shock to read this, and find none! At all!
Confess, you’re using spellcheck, aren’t you?
But not on your replies, so we have ‘The just aren’t instrumentat error.’ and ‘It will be interesting to look take a lok.’
Now, that’s more like it!
**I meant, instead of ‘wht you’re batting on about’, to put ‘what you’re ….’
of course.
SteveUK–
Spellcheck? What’s that?
I should use spellcheck, but I admit I don’t. Glad I could live up to my reputation. 🙂
Steve,UK-
You have now experienced an important internet law. It’s impossible to write a complaint about spelling and grammar without committing one yourself. 🙂
Lucia,
I agree with you. But in the end, that means that you don’t necessarily expect the “measurement error” to be white noise if it includes something else than instrumental error, especially if it includes arbitrary, but possibly biased, adjustments. I’m just trying to explain the result here.
FrancoisO
I agree that the measurement noise won’t be white if it’s influenced by biased adjustments. That’s true even if the bias is entirely accidental.
I’m showing this graph because I think it’s useful for people who think about the data to see the interesting shape of the correlogram! If it were ‘standard’ measurement noise, it should be fairly white. It’s not.
When I did an autocorrelation (GISS-HadCrut normalized to the same period for land ocean) in R using the acf function the peaks showed up at a yearly cycle. Would that imply a sinusoidal difference between the two data sets, since the acf of the lag of the difference is periodic? In other words might there be a seasonally based bias between the two data sets? Due to coverage perhaps?
Barry W–
Before the ‘anomaly’ process is imposed, there is a lot of energy in the annual cycle relative to actual “weather noise” and/or measurement uncertainty. So, I figure what ever we are getting at 12 months is just “left over”. It could be due to coverage, but it could just be a statistical feature related to exactly how each group decides what the “mean” temperature for January, Feb, March etc. is. I don’t know exactly how each one does it, but for the 30 year period I picked, if I take the (Giss-Hadley) difference by month, the average for January is not zero. Likewise for Feb. Etc.
It’s not a big difference, but it’s no zero either. That results in some “energy” at 12 months. I’m not sure it’s “due” to much of anything other than definitions of anomalies. (That said, it may be due to something.)
I’m not going to agonize it– just recognize it’s there. (Sometimes just knowing what’s there is as important as understanding why it’s there!)
Lucia,
Actually it’s an important finding. No one can claim that the uncertainty or error is solely “instrumental” error, and therefore white noise. The data are “contaminated” with something else. All this makes it extremely difficult to figure out what you would like to know: what is the actual “weather noise”? You now know that you can’t model the data as the sum of, say, “actual” temperatures, plus AR(1) weather noise and white instrumental noise. There is something else in there, and one must figure out what it is to include it in the model.