By Steve Goddard
We are all familiar with the GISS graph below, showing how the world has warmed since 1880.
http://data.giss.nasa.gov/gistemp/graphs/Fig.A2.lrg.gif
The GISS map below shows the geographic details of how they believe the planet has warmed. It uses 1200 km smoothing, a technique which allows them to generate data where they have none – based on the idea that temperatures don’t vary much over 1200 km. It seems “reasonable enough” to use the Monaco weather forecast to make picnic plans in Birmingham, England. Similarly we could assume that the weather and climate in Portland, Oregon can be inferred from that of Death Valley.
The map below uses 250 km smoothing, which allows us to see a little better where they actually have trend data from 1880-2009.
I took the two maps above, projected them on to a sphere representing the earth, and made them blink back and forth between 250 km and 1200 km smoothing. The Arctic is particularly impressive. GISS has determined that the Arctic is warming rapidly across vast distances where they have no 250 km data (pink.)
A way to prove there’s no data in the region for yourself is by using the GISTEMP Map locator at http://data.giss.nasa.gov/gistemp/station_data/
If we choose 90N 0E (North Pole) as the center point for finding nearby stations:
We find that the closest station from the North Pole is Alert, NWT, 834 km (518 miles) away. That’s about the distance from Montreal to Washington DC. Is the temperature data in Montreal valid for applying to Washington DC.?
Even worse, there’s no data in GISTEMP for Alert NWT since 1991. Funny though, you can get current data right now, today, from Weather Underground, right here. WUWT?
Here’s the METAR report for Alert, NWT from today
METAR CYLT 261900Z 31007KT 10SM OVC020 01/M00 A2967 RMK ST8 LAST OBS/NEXT 270600 UTC SLP051
The next closest GISTEMP station is Nord, ADS at 935 km (580 miles) away.
Most Arctic stations used in GISTEMP are 1000 km (621 miles) or more away from the North Pole. That is about the distance from Chicago to Atlanta. Again would you use climate records from Atlanta to gauge what is happening in Chicago?
Note the area between Svalbard and the North Pole in the globe below. There is no data in the 250 km 1880-2009 trend map indicating that region has warmed significantly, yet GISS 1200 km 1880-2009 has it warming 2-4° C. Same story for northern Greenland, the Beaufort Sea, etc. There’s a lot of holes in the polar data that has been interpolated.
The GISS Arctic (non) data has been widely misinterpreted. Below is a good example:
Monitoring Greenland’s melting
The ten warmest years since 1880 have all taken place within the 12-year period of 1997–2008, according to the NASA Goddard Institute for Space Studies (GISS) surface temperature analysis. The Arctic has been subject to exceptionally warm conditions and is showing an extraordinary response to increasing temperatures. The changes in polar ice have the potential to profoundly affect Earth’s climate; in 2007, sea-ice extent reached a historical minimum, as a consequence of warm and clear sky conditions.
If we look at the only two long-term stations which GISS does have in Greenland, it becomes clear that there has been nothing extraordinary or record breaking about the last 12 years (other than one probably errant data point.) The 1930s were warmer in Greenland.
Similarly, GISS has essentially no 250 km 1880-2009 data in the interior of Africa, yet has managed to generate a detailed profile across the entire continent for that same time period. In the process of doing this, they “disappeared” a cold spot in what is now Zimbabwe.
Same story for Asia.
Same story for South America. Note how they moved a cold area from Argentina to Bolivia, and created an imaginary hot spot in Brazil.
Pay no attention to that man behind the curtain.
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stevengoddard replied, “You are not making any sense. This map is the GISS 250km trend map… …Note the word ‘trend’ in the link.”
I always make sense.
Apparently, you forget that there is data associated with the maps. What the graphs in my earlier comment showed was, between the latitudes of 60S and 60N, the infilling of the 1200km radius smoothing has had basically no effect on the linear trends from January 1880 to June 2010. Read the comment and understand the graphs, Steven:
http://wattsupwiththat.com/2010/07/26/giss-swiss-cheese/#comment-441550
And to confirm my point, here’s another comparison graph. This one compares the GISTEMP combined land+sea surface temperature data with 250km and 1200km radius smoothing for one of your examples, Africa. The infilling of the 1200km radius smoothing added a whopping 0.002 deg C/decade to the linear trend for Africa, Steven.
http://i26.tinypic.com/25rp2fr.jpg
Ryan: July 28, 2010 at 6:26 am
Excuse me for throwing in my two cents. As I showed in my reply to Steven Goddard above…
http://wattsupwiththat.com/2010/07/26/giss-swiss-cheese/#comment-441550
…the infilling and smoothing caused by the 1200km radius smoothing has a significant impact only at high latitudes and this is caused in part by GISS deleting SST data and extending Land Surface data out over the oceans. Refer to:
http://bobtisdale.blogspot.com/2010/05/giss-deletes-arctic-and-southern-ocean.html
But between the latitudes of 60S-60N, the infilling and smoothing of the 1200 km radius smoothing has little to no effect:
http://i29.tinypic.com/55pc86.jpg
That’s the bottom line.
Gail Combs – your link contains some quite wrong statements about the Pearson correlation coefficient.
“1. Independent of case: In Pearson’s correlation of coefficient, cases should be independent to each other.”
Not so. This is one of the things we’re testing with correlations, isn’t it? If the two datasets were completely independent, there would be no correlation, would there?
2. Distribution: In Pearson’s correlation coefficient, variables of the correlation should be normally distributed.
Can’t see how this is relevant.
3. Cause and effect relationship: In Pearson’s correlation coefficient, there should be a cause and effect relationship between the correlation variables.
This contradicts point number one, and also the mantra of statistics: “correlation does not imply causation”. Two things can be strongly correlated without any causal connection at all.
4. Linear relationship: In Pearson’s correlation coefficient, two variables should be linearly related to each other, or if we plot the value of variables on a scatter diagram, it should yield a straight line.”
Not correct. If the relationship is non-linear, Pearson’s correlation coefficient is an underestimate of the true correlation, but that does not mean it can’t be used.
Generally, there is no reason not to use Pearson’s correlation coefficient to estimate the correlation between temperature anomalies.
An Inquirer: I think your comments were aimed at me rather than Gail Combs. A trend of 10% and a trend of 1% in two different anomaly series certainly would not imply that the two series were strongly correlated. Even if they were, the GISS methodology would do nothing that resembled “tossing away” the 1% and saying “the overall trend is 10%”. So I really don’t know what your point is.
It’s very easy to find two relatively nearby weather stations for which the anomalies do not correlate well. This is not news. If you look at the plots of this in Hansen’s paper which first described the situation, you can see very clearly that there is a lot of scatter about the relationship between correlation and distance. Nonetheless, the relationship clearly exists.
I started with January because that was the first column in the datasets I downloaded. The results for any other month, or the whole year, would be similar, and just as easy to compute. You’re welcome to replicate my analysis; I don’t have time.
Doubting Thomas:
Re: “same signal” when looking at surface energy rather than temperature.
All I meant by this was that you still see a clear warming signal when you look at surface enthalpy instead of surface temperature. But, as you say, when you look at enthalpy (which is a better reflection of energy in the air) the rate of warming is twice as fast ( at least according to that article). I’m not quite sure if we’re arguing or agreeing 😉
@Bob Tisdale: Basically you are throwing something into the heap that isn’t being discussed here at all. Its just plain obfuscation. Its also misleading. If you said that the Artic was melting and the Sahara was freezing then “on average” the trends would be the same. Not the same impact however, is it?
The point of these maps is that they show greatest warming at centers of greatest propaganda. They suggest that the Amazon is going to dry up and the Artic melt leaving the Polar bears with no home. However, there is not actual data to back up the claims made by these maps.
Every month, when GISTemp at:
http://data.giss.nasa.gov/gistemp/tabledata/GLB.Ts+dSST.txt
…. is updated, there are over 100 changes in previous monthly averages in their record back to 1880.
Last month, 6 of the 12 monthly numbers from 1880 were changed. 8 of the 12 months from 1881 were changed and so on.
Naturally, the majority of the earlier records are adjusted down and the majority of the recent records are adjusted up. The switch date seems to be around 1977.
This happens every month. I have been tracking it for over two years with this useful website.
http://www.changedetection.com/
So let’s say one has been doing this for 20 years. Each month, you bump down half of the 1880’s by a -0.01C, 20 years * -0.005C average * 12 months = -1.2C [ie. there is lots of opportunity to make a major change in the trend.]
Let’s compare the recent GISTemp to the earliest version at the Wayback Machine – 2005 – 1880 is now 0.06C cooler than it was before (which seems to be about the average for all the 1880s), 2003 is is now 0.03 warmer than it used to be. So 5 years of changes have added 0.1C to the trend alone.
http://web.archive.org/web/20050914112121/http://data.giss.nasa.gov/gistemp/tabledata/GLB.Ts+dSST.txt
Ryan replied, “Basically you are throwing something into the heap that isn’t being discussed here at all.”
Then disregrard what I wrote. Simple as that.
Bill Illis wrote
“Last month, 6 of the 12 monthly numbers from 1880 were changed. 8 of the 12 months from 1881 were changed and so on. ”
How are these “adjustments” justified?
In engineering, changing old data is called fudging the data or cheating!
What reason, other than some new data like a thermometer calibration or time of observation can be used to validly adjust data. And why would adjustment s be needed 130 years later, surely diligent workers would have caught this stuff in the 1980s!
“Bill Illis says:
July 28, 2010 at 9:09 am
Every month, when GISTemp at:
… is updated, there are over 100 changes in previous monthly averages in their record back to 1880.”
Interesting observation. This could have something to do with how the model regenerates every month while keeping the base period (1951-1980) as the zero point? Another one against anomalies….
Tisdale wrote:
But between the latitudes of 60S-60N, the infilling and smoothing of the 1200 km radius smoothing has little to no effect:
http://i29.tinypic.com/55pc86.jpg
Bob, read this very carefully please.
Set 1, real data
(10+20+30+40+50+60+70+80+90+100)
Set 2, infilled by interpolation of indexes
(10+15+20+25+30+35+40+45+50+55+60+65+70+75+80+85+90+95+100)
Amazingly, the mean of both sets is 55. My ‘infilling’ with averages had no effect on my outcome.
The reason they are both 55 is because the additional data in set 2 comes from interpolation of data set 1. I have added a data set with the same mean to a data set to produce a larger data set with the same mean, but new statistical characteristics.
Data set 2 has 9 more elements than data set one. From a layman standpoint, I have more confidence in my average because I have used more data. From a mathematical standpoint, I have produced recursive, meaningless indexes to the mean but still lowered my standard deviation (commonly used for mathematical error calculations.)
However, critically, from a scientific standpoint, I have produced additional data points which have double the pure error of each index from which it is computed. I have not reduced the pure error of my mean at all.
So, making conclusions based on 250km OMITS the empty space from which you are not taking data. Making conclusions based on 1200km smoothing created additional data coverage which has not improved your error against reality. These are mathematical tricks.
In reality, at 250km smoothing, I leave about, say, 90% of Africa untouched. Therefore, at 250km smoothing, computing the average temperature of Africa, I have 10% accuracy by area. At 1200km smoothing, I have now covered 90% of Africa, 80% of that being manufactured data. I have 10% accuracy.
If the data doesn’t exist at smaller smoothing, it shouldn’t be used at higher smoothing unless the system is proven to be linearly coupled (IE the average between any two points of equal distance is invariant to the sample size between those distances, provided the data set is symmetrically distributed) Prove that and you win the argument…
Mosher:
Anomaly measures the change AT A LOCATION from its own local mean.
…
its NOT TEMPERATURE Changes over 1200km. Its the change in temperatures VERSUS their local mean. You can think of this as a trend change
So.. what you are saying is..
TO BE ACCURATE, EACH RECORD MUST HAVE AN ACCURATE LOCAL MEAN
TO BE ACCURATE, EACH MEAN MUST INCLUDE THE WHOLE SAMPLE PERIOD
TO BE ACCURATE, EACH RECORD MUST BE CONTINUOUS OVER THE PERIOD
TO BE ACCURATE, EACH MEAN IS LOCAL: NOT INTERPOLATED
THEREFORE, INTERPOLATED MEANS TO DERIVE INTERPOLATED ANOMALY IS FAKE DATA
How accurate is a station 1 kilometer away? Giving each area of measurement 1200km of effect means you are assuming that it reflects accurately the local mean OVER 1200KM. That is pure crap.
These are math tricks, and they are wrong. End of story.
Buffoon wrote, “Bob, read this very carefully please,” and you went into a wonderful description of impacts of interpolation. You concluded with, “If the data doesn’t exist at smaller smoothing, it shouldn’t be used at higher smoothing unless the system is proven to be linearly coupled.”
Actually, I object to the use of the 1200km radius smoothing for another reason: the illusion of a spatially complete instrument temperature record. The 1200km radius smoothing gives a false impression of the instrument temperature record. It portrays that the data is complete globally from 1800 to 2010, when it is far from complete. Refer to:
http://bobtisdale.blogspot.com/2010/01/illusions-of-instrument-temperature.html
The point of my exercise on this thread, though, was to show something entirely different. There is a common belief that the GISS 1200km radius smoothing raises the trend in temperatures globally, and I have shown that this is not the case. As you quoted [and I will clarify with an addition], But between the latitudes of 60S-60N, the infilling and smoothing of the 1200 km radius smoothing has little to no effect [on long-term trends]:
http://i29.tinypic.com/55pc86.jpg
Regards
What Bob Tisdale said
Bob,
If that is the case, I simply did not manage to figure out which side of the issue you were arguing. In such an event, apologies for my (correct but not necessarily warranted) pedantics. Let us join hands and run merrily through interpolated fields.
“But between the latitudes of 60S-60N, the infilling and smoothing of the 1200 km radius smoothing has little to no effect [on long-term trends]:”
I see what you are saying, but, I suspect my argument may be, without the 1200km smoothing, we do not reach the threshold for spatial coverage from which we may have GISS’ 66% magic number. At 250km there is an abhorent lack of spatial coverage, at 1200km there is fake spatial coverage which won’t change the 250km trend since it’s derived from the same dataset.
Steve’s original post seems to be : Spatially adequate data for the period doesn’t exist for the whole period and thus there can’t be a valid trend. I wholeheartedly support that conclusion, suggesting that interpolated data provides enough spatial coverage to subjectively meet some threshold value, but the underlying dataset is not complete enough to support the conclusion that is the only valuable purpose of the first graph on the page
Are we, in fact, on different sides of the issue or not?
Jay and Jose Suro: Some of the changes in the GISTEMP product result from data arriving late or dues to corrections in the supplied data. And GISS makes improvements to its product from time to time and these changes can have a wide range of small effects, and impact early data. Refer to:
http://data.giss.nasa.gov/gistemp/updates/
And then someone will notice a shift in the value of one month early in the record and there’s no explanation.
Lucia tries to monitor the changes.
http://rankexploits.com/musings/
Bob Tisdale says:
July 28, 2010 at 1:04 pm
The point of my exercise on this thread, though, was to show something entirely different. There is a common belief that the GISS 1200km radius smoothing raises the trend in temperatures globally, and I have shown that this is not the case.
This only (as does the example given by Buffoon) only when the “smoothing” smooths. Smoothing does not, for example, create hot spots in Brazil.
A more accurate representation of what is seeming to happen is if we take sequence two from Buffoons example and try to transform it into sequence one by dropping the interpolations. Except we don’t drop all the interpolations at once, we drop them one at a time, starting with the lower numbers. Let’s call this something snappy like “progress” – let the critics call it something snappier like: “The March of the Thermometers”. What happens to the average over time then? Who thinks we’re at the end of this process?
RW says:
July 28, 2010 at 8:18 am
Gail Combs – your link contains some quite wrong statements about the Pearson correlation coefficient….
2. Distribution: In Pearson’s correlation coefficient, variables of the correlation should be normally distributed.
Can’t see how this is relevant.
_________________________________________________________________
The statement you made that the data must be normally distributed is irrelevant. Is exactly what I was talking about.
People with very little knowledge of statistics dump numbers into a stat program and think that just because they got numbers out everything is OK and they are proving something.
The whole blasted mathematical basis upon which Pearson correlation coefficient rests is based on NORMALLY DISTRIBUTED DATA. IF it AIN”T NORMALLY DISTRIBUTED it is garbage in and garbage out.
bemused says:
July 27, 2010 at 4:41 pm
MarkG:
“How can anyone claim ‘robust correlation’ between widely separated areas when two stations only a short distance apart show a difference of 2.5C for pretty much an entire decade?”
…Sure, there are a few station pairs which have low correlations even though they are close together, but the vast majority of stations show a good correlation.
Everyday experience also tells you this -if the summer was colder than average in Boston and Atlanta, then it was probably colder than average in Washington DC as well.
________________________________________________________________
OK than we will go with your colder than average. It was freeze you tail off cold in North Carolina, Florida, England, Spain and so cold in Tibet it killed more than a million animals. Even the Russians and Chinese were complaining of the cold this winter. The USA has snow in ALL fifty states.
Here in North Carolina it was colder than normal until the middle of June and the people in California and Oregon are still complaining of the cold. South Africa has dead penguins and South America dead cows because of the cold. The Aussies are also saying it is cooler than normal. Even the sea temperatures have turned cold. So if this world wide temperature “correlation” is so “scientifically” correct how come we are hearing “it is the hottest year EVAHHhh” ????
Z: You replied, “This only (as does the example given by Buffoon) only when the ‘smoothing’ smooths. Smoothing does not, for example, create hot spots in Brazil.”
I can’t describe or comment on the creation “hotspots in Brazil” because I haven’t investigated it. But GISS extrapolates data. It is not simply a smoothing process.
And please understand that the sentence you are quoting was written because I was illustrating that, outside of the high latitudes, the 1200km radius smoothing does not add to the long-term temperature anomaly trend. As I noted to Buffoon, I am not defending the use of the 1200km radius smoothing. In fact, I’m against its use because it gives the false impression of a globally complete instument temperature record from 1880 to present, when, in reality, the record is far from complete.
Buffoon wrote, “Steve’s original post seems to be : Spatially adequate data for the period doesn’t exist for the whole period and thus there can’t be a valid trend.”
If you were to run through the thread, you’d find that Steve’s opinion is based on the lack of data in the end years of the data with 250km radius smoothing. Refer to his July 27, 2010 at 5:41 pm comment:
http://wattsupwiththat.com/2010/07/26/giss-swiss-cheese/#comment-441283
Tisdale wrote:
But between the latitudes of 60S-60N, the infilling and smoothing of the 1200 km radius smoothing has little to no effect:
http://i29.tinypic.com/55pc86.jpg
Bob, read this very carefully please.
Set 1, real data
(10+20+30+40+50+60+70+80+90+100)
Set 2, infilled by interpolation of indexes
(10+15+20+25+30+35+40+45+50+55+60+65+70+75+80+85+90+95+100)
Simple demonstration of the Nyquist sampling theorem:
You have the following real data set on a grid of say 80 km:
(1, 1, 1, 1, 2, 0, 1, 1, 1, 10, 1, 1, 1, 1, -5)
15 real datapoints spaced over 1200 km.
But because of sparsity of measerment stations, you know only the number a index 0 (1), index 9 (10) and index 14 (-5). Linear interpolation (smoothing) would infill the missing ones, creating the following set:
(1,2,3,4,5,6,7,8,9,10,7,4,1,-2,-5)
The average of the interpolated data set is 4, while the average of the real data is 1.2.
Gail Coombes wrote:
“OK than we will go with your colder than average. It was freeze you tail off cold in North Carolina, Florida, England, Spain and so cold in Tibet it killed more than a million animals. Even the Russians and Chinese were complaining of the cold this winter. The USA has snow in ALL fifty states.
Here in North Carolina it was colder than normal until the middle of June and the people in California and Oregon are still complaining of the cold. South Africa has dead penguins and South America dead cows because of the cold. The Aussies are also saying it is cooler than normal. Even the sea temperatures have turned cold. So if this world wide temperature “correlation” is so “scientifically” correct how come we are hearing “it is the hottest year EVAHHhh” ????”
Hi Gail, maybe it has something to do with the fact that globally it was a very warm period between December 2009 – February 2010. Sure it was cold over the population centers of Europe and the US, but that is not global.
Did you know that USA takes up only 1.9% of the global surface area? United Kingdom takes up only 0.05%, Australia takes up 1.5%, China also takes up 1.9%, Russia takes up 3.3%, South America is 3.5% (those are the places you mentioned). All those combined only come to just over 12% of the global surface area.
Nobody is saying that correlations are global, only out to around 1000km. Did you know that a 1200km diameter circle (the approximate area where GISS assumes some correlation) occupies only 0.2% of the global surface area – that is hardly a global correlation. So, just because it was a cold winter in some places doesn’t mean that it must have been cold globally.
If you don’t want to take my work for it, take it from Roy Spencer. His satellite data show it to have been a very warm year (particularly last winter -N Hemisphere):
http://discover.itsc.uah.edu/amsutemps/execute.csh?amsutemps
http://www.drroyspencer.com/2010/07/june-2010-uah-global-temperature-update-0-44-deg-c/
You can argue over whether or not 1998 or 2010 will end up being the warmest year globally since we began taking measurements, but you cannot deny that the last 12 months have been among the warmest few. Just because you had a bit of snow in your back garden doesn’t change that. If you don’t trust the satellite data then I suggest you take that up with Roy Spencer.
bemused says:
July 28, 2010 at 5:05 pm
Hi Gail, maybe it has something to do with the fact that globally it was a very warm period between December 2009 – February 2010. Sure it was cold over the population centers of Europe and the US, but that is not global.
Nobody is saying that correlations are global, only out to around 1000km. Did you know that a 1200km diameter circle (the approximate area where GISS assumes some correlation) occupies only 0.2% of the global surface area – that is hardly a global correlation. So, just because it was a cold winter in some places doesn’t mean that it must have been cold globally.
_________________________________________________
I am well aware that the January, February and March data was warmer because the sea temperatures were warm and the oceans make up 70% of the earth’s surface. However I have a major problem with the Climate Scientists playing fast and loose with statistics. I am no Steve M, but I made my living as a quality engineer using statistics on a regular basis so I cringe when I see mismatched data blythely homongenized.
I live in the piedmont/sand hills area. The weather patterns are not the same as in the mountains or the other side of the mountains (500 km) or the seashore (250 km) as I demonstrated in my comment:
Gail Combs says:
July 27, 2010 at 2:02 pm
If you want to do the type of “homogenizing” that is done you had better do the correct type of sampling AND take into account the fact that terrain changes weather patterns. Also do not expect me to believe the data is good to +/-0.1C
I have written to many sampling plans for liquids, solids and widgets to believe what climate scientists are trying to pedal.
Gail Combs:
“People with very little knowledge of statistics dump numbers into a stat program and think that just because they got numbers out everything is OK and they are proving something.”
It seems to me that you have very little knowledge of statistics. You found a web page that made a number of mistaken claims about the Pearson coefficient, and you seem to think that permits you to make statements like this. The web page you found was wrong; those four statements about the Pearson coefficient are wrong.
“The whole blasted mathematical basis upon which Pearson correlation coefficient rests is based on NORMALLY DISTRIBUTED DATA. IF it AIN”T NORMALLY DISTRIBUTED it is garbage in and garbage out.”
This is not correct at all. The distribution of the data is to a large extent irrelevant. If you compare the data sets (1,2,3) and (10,20,30), you will find that neither is normally distributed, the Pearson coefficient will be 1, and this makes perfect sense. All the Pearson coefficient is doing is telling you on a scale of 0 to 1 how well a linear relationship can explain the covariance of two variables. The mathematical definition of the Pearson coefficient does not in fact require any particular statistical distribution of the values of the two variables.
It’s poor form to throw around phrases like “People with very little knowledge of statistics…”, even if you yourself understand statistics. It’s much worse when you’re making basic errors.
Bob
If you were to run through the thread, you’d find that Steve’s opinion is based on the *lack of data* in the end years of the data
//Lack of data means the trend isn’t valid. Are you saying there isn’t a lack of data? I don’t get what you’re arguing.
KLA
(1, 1, 1, 1, 2, 0, 1, 1, 1, 10, 1, 1, 1, 1, -5)
15 real datapoints spaced over 1200 km.
But because of sparsity of measerment stations, you know only the number a index 0 (1), index 9 (10) and index 14 (-5). Linear interpolation (smoothing) would infill the missing ones, creating the following set:
(1,2,3,4,5,6,7,8,9,10,7,4,1,-2,-5)
//Sorry, I don’t follow. You give me a set of real data with 15 indexes. Then you construct an interpolation example from 3 indexes. If you interpolate over real data, you can calculate error by location. The average error in your interpolated set vs. your real set is 2.8, handily, exactly the difference in means. You can’t make free accuracy.
Nyquist sampling says that a signal can be reconstructed perfectly if the sampling rate is 2 times the highest frequency in the data It therefore assumes the data is functional. Temperature data as taken is not functional, it is stepped across irregular areas. The void is assuming that the temperature data is linearly coupled such that it would be subject to the sampling laws for bandlimited signals or averaging processes. Please demonstrate the sampling frequency required for accurate temperature reconstruction of, say, Africa, by finding the smallest distance over which there can be a significant step change, and then come back to me with the suggestion that the data sampling rate shown in the 250km graph would support nyquist-shannon reconstruction with an expectation of accuracy.