Sep 08 2008
Instrument Error in the Global Mean Temperature Anomaly
It’s that time of month again. The monthly temperature anomalies are soon to be publicized. I’d usually ignore them, but I’ve had this (simple) script hanging around for about a month now, so I thought I’d use it. With UAH releasing their August temperature datum, and temperatures plunging in 2008 it seemed like a good opportunity to look at what the monthly changes in temperature anomaly mean (or don’t mean).
What this post will attempt to do is show the effects of instrument noise on the global mean temperature anomaly time series. I’m going to assume that anyone reading this has seen at least one of the temperature anomaly time series. Pick your favorite - it doesn’t really matter which one. I chose the GISTemp anomaly from 1880.
Instrument or Weather Noise
The procedure is quite simple. Find the temperature at time index i and compare it to the temperature at time index i-1; dt = t[i]-t[i-1]. Thus if the temperature has decreased in the last month, dt will be negative. This is done for all values of i in the GISTemp record.
The results are shown below:
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It is clear that this is nearly normal (Gaussian). I’ve included the mean and standard deviation on the graph. The mean is 0.00047 and the standard deviation is 0.12600. Rounding to 2 decimal places, this means that the change in temperature from one month to the next is 0.00+/-0.13. From one month to the next, there is not a large change in the global mean surface temperature.
The question is whether this is “weather noise” or “instrument noise”. Because of its near-normal shape, I argue that it is instrument noise. If there were no weather on the Earth, there would still be some measurement error in the temperature data due only to the uncertainty in making a measurement.
Consider a physics101 experiment. Climb to the roof of a building, and drop a tennis ball 100 times. Record the amount of time it takes for the ball to drop using a stopwatch. All of the measurements will not be alike. This is the “noise” or “error” due to the instrumentation that we used. If you use a more sophisticated technique of measuring time you can reduce the error but it can never be eliminated.
Made Up Data
We can see how this instrument noise might manifest itself in the temperature record by looking at hypothetical data. I’ve used the same procedure to look at data with a trend and white noise.
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This did not use as many points in the time series, so the histogram doesn’t appear near as normal. Even so, it is remarkably similar to the GISTemp histogram above. The amount of white noise in the plot was designed so that it had the same characteristics as the GISTemp noise. The standard deviation of the instrument noise was 0.08C.
When the trendy-noise is plotted, the large amount of noise relative to the trend can be seen. However, this is not the same as the total amount of noise in the temperature data. I did not account for “weather noise” - so called red noise. ENSO is an example of red noise.
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I’ve plotted the latest 4 year trend in this hypothetical data. Notice that it is negative even though the prescribed trend is quite positive. I admit, I did go searching for this, but not as long as I originally thought I would. It is to show that negative 4 year trends can be the result of only instrument (white) noise; there is no need to invoke weather noise (red).
For comparison, I’ve plotted my fake time series with only a trend and white noise in black and one of the global mean temperature anomalies in red. This should show that there is both red and white noise in the global temperature time series.
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6 Responses to “Instrument Error in the Global Mean Temperature Anomaly”
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If that 0.13C error associated with the change in temperature from one month to the next is really due to “instrument noise”, as you say, then we should be able to estimate (roughly) the number of individual instrumental measurements that went into each monthly average.
First, if the uncertainty associated with a single instrument (thermometer) measurement in this case is +- 1 degree C (probably not a bad first assumption), a rough estimate of the uncertainty associated with the mean of 100 such measurements might be +- 1/10 deg (ie, 1 divided by the square root of the number of individual measurements “N” that were averaged together.)
Second, If the uncertainty in the monthly average temp at time t is “+- Y” deg and that in the monthly average temp at time t+1 (ie, next month) is also “+- Y”, then the uncertainty in the calculated change from the first month to the next should be roughly “+- Y times square_root_of (2)” deg (ie, the uncertainty associated with the average temp for any given month multiplied by the square root of 2.)
So, let’s assume each instrumental (ie, thermometer) temperature measurement at any given temperature station has an associated uncertainty of +-1 degree C and also use the +-0.13C number you give above as the uncertainty for the difference in temp from one month to the next. That means the uncertainty in the mean value for any given month is actually about +- 0.09C. (ie, 0.13C divided by the square root of 2)
so, from the above assumptions we can estimate the approximate number of measurements averaged together to come up with mean
1 / square_root(N) = 0.09
or N = 123
Is that right?
Each GISS monthly average was calculated using only 123 individual temperature measurements?
I thought that the GISS monthly average was calculated from a relatively large number of individual measurements (larger than 123, at any rate).
If we assume that two measurements are performed per 24 hour period at each station ( min and max. for example) and that measurements are taken on each of 30 days for 1 month, that is already 60 individual measurements for just one station.
123/ 60 is equal to about 2
Temperature data from just 2 stations was used to get the monthly average???
Am I missing something here?
Jim–
There measurement noise in the data comes from more than just thermometry errors. Lack of station coverage is important. GMST is supposed ot be for the entire earth, but the earth is only sampled at a finite number of locations. Since the anomaly over the surface of the earth, the measurement error is larger than you would estimate based on that calculation.
It’s also not clear the errors from month to month are white noise. Since weather patterns persist, there could be some correlation over time in the errors.
[Reply: You're right. Month-to-month variations would not be only white noise. It would be a combination of white and red noise.]
Atmoz–
The weather observations do look like they could be white noise superimposed on red noise. How did you select 0.8C for the standard deviation for measurement error? I’ve the paper discussing Hadley’s estimates for their data, tried to compare system to system and tried to do a few reality checks. So, I’m curious to know how you decided on your value. Thanks.
[Reply: Guess and check. But I suppose histogram matching sounds a little more scientific. The mean and standard deviation of the histograms don't exactly match in the 2 figures because, as I stated, I was fishing for the negative slope in the last 4 years - and I wanted to limit the total length of the time series to ~28 years. It wasn't a totally quantitative method. As soon as I got close to the right values, I said "that looks close enough for a blog post". I welcome further comments and criticisms of the methodology.
Which Hadley paper are you referring to? I don't think I've read it. If so, I don't remember it.]
Oh– instead of just looking at time series, a better way to show there is both red and white noise in the temperature series is the correlogram. Plot the log of the correlation vs. the lag. If you skip the autocorrelation at zero lag the data will fall on a straight line, but the intercept won’t go through ln(1), it will go through a lower value.
If you generate the really series of red noise (for simulated weather) and white (for instrument error) and create the correlogram, you’ll see this actually works. But it can be shown with about two lines of math too.
Thanks. I guess I was thinking too narrowly when I saw the term “instrument noise” in the original post.
So my next question is, just how important is the incomplete station coverage?
If one assumes that the error in the monthly mean anomaly does not include “weather noise” (the assumption of the original post), it would seem that the latter effect would have to be important indeed.
I don’t know the precise number of temperature measurement stations used by GISS (and I realize it has not even remained constant over the years), but I believe it is on the order of 1000.
Based on my calculation from my above post, that would imply that incomplete coverage has the result of reducing the effective number of measurement stations from 1000 to 2 (!)
But, it appears that both of you (Lucia and Atmoz) now believe that the error in the monthly mean anomaly is actually due to more than just measurement error, at any rate.
Based on my above (admittedly simple) analysis, I have the same hunch.
My above use of
in a couple places should instead read
Ie,
it appears that both of you (Lucia and Atmoz) now believe that the error in the change in monthly mean anomaly is actually due to more than just measurement error, at any rate.
Atmoz,
Hadley cites their own paper here: http://hadobs.metoffice.com/hadcrut3/index.html
Use your browser search tool to find “references”.
I don’t have any criticisms! I just did some fiddling to estimate the same number, and I can’t get a stupendously defensible number!
Jim–
Yes, I do think the changes from month to month not entirely do to error! However, one reason I’m asking Atmoz about his estimate is that if the “uncertainty” has a standard deviation of 0.08C, that would suggest that a very high fraction of the month-to-month change in the recent period is measurement uncertainty! ( The residuals to a linear fit for Hadley are about 0.09C= 0.10C for the past 90 months! So, 0.08C would be a very large fraction of that.)