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For the second statistical test I will again focus on HadCRUT, GISTEMP, BEST, and ERA since these are all surface datasets that have full sphere converge. This test will compare the difference between monthly measurements of each dataset and the mean of all. The mean is considered to be the best expectation of the true value. The quantity we are calculating is Dx = Tx - Tavg where Dx is the difference between the temperature Tx of dataset x and the average temperature Tavg. The period of evaluation is 1979/01 to 2021/12 which covers 516 months. That means each dataset x has 516 measurements than be compared to the average. We will determine the uncertainty of Dx as 2*u(Dx) by calculating the standard deviation of Dx. Since it is common practice to report uncertainty at 95% confidence we will multiple by 2 for the 2-sigma range. Again, this a type A evaluation of uncertainty. For HadCRUT this value is 0.051 C, for Berkeley Earth it is 0.060 C, for GISTEMP it is 0.057 C, and for ERA it is 0.086 C. The average implied uncertainty is 0.065 C. I am a little surprised by ERA. It has the highest uncertainty wrt to the mean of the datasets analyzed. I'm surprised because ERA is considered to be among the best reanalysis datasets and incorporates not only orders of magnitude more observations than the other datasets but many different kinds of observations including surface, satellite, radiosonde, etc. It has a much longer tails on the distribution. So we've calculated average implied uncertainty 2u(D) as 0.065 C. But that is only the uncertainty of the difference wrt to the average. The average itself will have an uncertainty given by u(avg) = u(x) / sqrt(N). So u(Tavg) = u(D) / sqrt(N) = 0.0325 / sqrt(4) = 0.0163 C. So we have u(D) = 0.033 and u(Tavg) = 0.0163. We will apply the root sum square rule to find the final uncertainty u(T). It is u(T) = sqrt(u(D)^2 + u(Tavg)^2) = sqrt(0.033^2 + 0.0163^2) = 0.0368 C. And using the 2-sigma convention we have 2u(T) = 0.0368 * 2 = 0.074 C. That's our final answer. Method 1 above yields u = 0.075 C while method 2 here yields u = 0.074 C. This isn't much of a surprise that both methods give essentially the same result since they are both calculated from the same data.

There is no way 1.5 C is realistic anymore. Even at a modest sensitivity of 0.5 C per W/m2 that is 1.2 W/m2 * 0.5 C/W.m2 = 0.6 C of warming already queued up and in the pipeline to be added to the ~1.1 C of warming that has already occurred. And the evidence suggests the climate sensitivity in C per W/m2 may start low and increase as more tipping points are activated. A final 1 C per W/m2 or higher is very much in the realm of possibility still.

For the first statistical test of the uncertainty I focused only on HadCRUT, GISTEMP, BEST, and ERA since these are all surface datasets that have full sphere converge. In other words, they are all measuring almost exactly the same thing. This test will determine the difference between any two measurements. The quantity we are calculating is D = Ta - Tb where D is the difference and both Ta and Tb are temperature anomalies for the same month from randomly selected datasets. The period of evaluation is 1979/01 to 2021/12 which covers 516 months. With 4 datasets there are 6 combinations. This gives us a total of 3096 comparisons or values for Ta, Tb, and D. What we want to evaluate first is the uncertainty in the difference u(D). This is pretty simple since for type A evaluations it is the standard deviation of all values of D. In the graphic below you can see the histogram of the difference. The distribution is pretty close to normal and has a standard deviation of 0.053 C. So we set u(D) = 0.053 C. We're not done yet though. We know u(D) = 0.053 C, but what we really want to know is u(T). We can easily do this via the well known root sum square or summation in quadrature rule which says u(z) = sqrt[u(x)^2 + u(y)^2] for a function f in the form f(x, y) = x + y. The more fundamental concept that applies to any arbitrarily complex function f is the partial differential method, but there is no need to apply the more complex general form since our function f(x, y) = x + y is simple and is already known to propagate uncertainty via u(z) = sqrt[u(x)^2 + u(y)^2]. Assuming u(Ta) and u(Tb) are the same and can be represented with just u(T) the formula becomes u(D) = sqrt[2 * u(T)^2]. Solving this equation for u(T) we get u(T) = sqrt[u(D)^2 / 2]. So if u(D) = 0.053 then u(T) = 0.038 C. And our 2σ expanded uncertainty is 2*u(T) = 2 * 0.038 = 0.075 C. That's the result. The expanded 2σ uncertainty for monthly global average temperature anomalies using a type A evaluation where we compare each dataset to the others yields is 0.075 C. Note that is only one among multiple different ways a type A evaluation can be performed.

I have downloaded the following global average temperature products from several datasets. This datasets include 4 surface, 2 satellite, 1 radiosonde, and 1 reanalysis. UAHv6 - Satellite - Spencer et al. 2016 Data: https://www.nsstc.uah.edu/data/msu/v6.0/tlt/tltglhmam_6.0.txt RSSv4 - Satellite - Mears & Wentz 2017 Data: https://data.remss.com/msu/monthly_time_series/RSS_Monthly_MSU_AMSU_Channel_TLT_Anomalies_Land_and_Ocean_v04_0.txt RATPAC 850-300mb - Radiosonde - Free et al. 2005 Data: https://www.ncei.noaa.gov/pub/data/ratpac/ratpac-a/ NOAAGlobalTemp v5 - Surface - Haung et al. 2020 Data: https://www.ncei.noaa.gov/data/noaa-global-surface-temperature/v5/access/timeseries/ GISTEMP v4 - Surface - Lenssen et al. 2019 Data: https://data.giss.nasa.gov/gistemp/graphs_v4/graph_data/Monthly_Mean_Global_Surface_Temperature/graph.txt BEST - Surface - Rhode et al. 2013 Data: http://berkeleyearth.lbl.gov/auto/Global/Land_and_Ocean_complete.txt HadCRUTv5 - Surface - Morice et al. 2020 Data: https://www.metoffice.gov.uk/hadobs/hadcrut5/data/current/download.html ERA5 (Copernicus) - Reanalysis - Hersbach et al. 2020 Data: https://climate.copernicus.eu/surface-air-temperature-maps All datasets have been processed a common baseline: the full range 1979-2021 average. This is done so that they can be compared with each other. It should be noted that it is my understanding that of the surface datasets NOAAGlobalTemp v5 is still only a partial sphere estimate. It appears research is underway to make it full sphere. See Vose et al. 2021 for details. I do not believe this is publicly available yet. One criticism I often see is that the global average temperature warming trend is being overestimated. To test this validity of this claim I compared each of the above datasets. I also formed an equal weighted composite of the datasets for comparison. It is important to note that the dataset do not all measure the same thing. UAH and RSS measure the lower troposphere and do so at different average heights. RATPAC is the average from 850mb to 300mb which I selected to be representative of the UAH and RSS depths although neither UAH nor RSS are equally weighted in the 850-300mb layer. And as mentioned above NOAAGlobalTemp is a partial sphere estimate while GISTEMP, BEST, HadCRUT, and ERA are full sphere. In that context it could be argued that forming a composite is inappropriate. However, I do so here because all of these datasets are or have been used as proxy for comparisons of the "global average temperature" whether it was appropriate to do so or not. In the graph below I have plotted each dataset including the 13 month centered average of the composite. The formula used Σ[Σ[Tdm, 1, m], 1, d] / (dm) where d is the number of datasets (8) and m is the number of months (13). The composite 2σ timeseries represents an implied uncertainty based off the standard deviation from the 13 month centered average of the 8 datasets that go into it. The formula used is sqrt(Σ[(Tdm - Tavg)^2, 1, dm] / (dm - 1)) * 2 which is the formula for standard deviation multiplied by 2. It is primarily confined to a range of 0.05 to 0.10 C. However, it is important to note that UAH and RSS add a considerable amount of variance to the composite. A 13 month average would be expected to have a lower uncertainty than a 1 month average. That is true in this case as well though it may be hard for the astute read to see since typical monthly uncertainties for HadCRUT, GISTEMP, and BEST are generally on the order of 0.05 C. One thing that is immediately obvious is that UAH is, by far, the low outlier with a warming trend of only +0.135 C/decade. This compares with the composite of +0.187 C/decade. Also note the large difference between UAH and RATPAC and the small difference between RSS and RATPAC. It is often claimed that UAH is a better match to the radiosonde data. This could not be further from the truth at least when comparing with RATPAC which contains the homogenization adjustments making it valid climatic trend analysis. What I find most remarkable about this graph is the broad agreement both in terms of long term warming and the short term variation even though the methods of measuring the global average temperature use wildly different methodologies and subsets of available data.

We're probably getting pretty close to the Antarctica minimum. The NSIDC 5-day average is at 1.929e6 km2 as of Feb. 24th.

Because the Frank 2010 paper is still being promoted by the author and making its rounds in the blogosphere I thought I would dedicate an entire post to it. The publication claims that the lower bound of the annual global average temperature uncertainty is ±0.46 C (1σ) or 0.92 C (2σ). This result is then used by the blogosphere to conclude that we do not know whether or not the planet has warmed. Before I explain what I think is wrong with the Frank 2010 publication I'll first refer readers to the rigorous uncertainty analysis provided by Berkeley Earth via Rhode et al. 2013, GISTEMP via Lenssen et al. 2019, and HadCRUT via Morice et al. 2020. Each of these dataset unequivocally show that the planet is warming at rate of about +0.19 C/decade since 1979. And each uncertainty analysis confirms that the true uncertainty on the monthly values is about ± 0.03 (1σ) despite using wildly different techniques. Berkeley Earth uses jackknife resampling, GISTEMP uses a bottom-up type B evaluation, and HadCRUT uses ensembling not unlike a monte carlo simulation. The entirety of the conclusion from the Frank 2010 publication boils down to this series of calculations. (1a) σ = 0.2 "standard error" from Folland 2001 (1b) sqrt(N * 0.2^2 / (N-1)) = 0.200 where N is the number of observations (2 for daily, 60 for monthly, etc.) (1c) sqrt(0.200^2 + 0.200^2) = 0.283 (2a) σ = 0.254 gaussian fit based on Hubbard 2002 (2b) sqrt(N * 0.254^2 / (N-1)) = 0.254 where N is the number of observations (2 for daily, 60 for monthly, etc.) (2c) sqrt(0.254^2 + 0.254^2) = 0.359 (3) sqrt(0 283^2 + 0 359^2) = 0.46 Explanation of above: (1b) and (2b) are an attempt to propagate Tmax and Tmin uncertainties into a 30yr average used as the anomaly baseline and for an annual average. (1c) and (2c) is the propagation of uncertainty for an annual anomaly value. (3) is the combined uncertainty of the Folland and Hubbard components after propagating into anomaly values. Here are the 3 mistakes I believe the author made in order of increasing egregiousness. These are based on my direct conversations with the author. Even the first mistake is egregious enough to that it would get rejected by a reputable peer reviewed journal. The other 2 are so egregious it defies credulity that it even made it into the Energy & Environment journal which is actually more of a social and policy journal than a science journal and has a history of publishing research known for being wrong. Mistake 1. The uncertainties provided by Folland 2001 and Hubbard 2002 are for daily Tmax and Tmin observations. The author combines these in calculation (3) via the well known root sum square or summation in quadrature rule under the assumption that Folland and Hubbard are describing two different kinds of uncertainty that must be combined. The problem is that Folland is terse on details. It is impossible to say exactly what that 0.2 figure actually means. But based on context clues I personally inferred that Folland is describing the same thing as Hubbard. They just came up with slightly different estimates of the uncertainty. Mistake 2. The formula used in steps (1b) and (2b) is σ_avg = sqrt[N * σ^2 / (N-1)] where N is the number of the observations included in a time average and σ is the daily Tmax or Tmin uncertainty. For example for a monthly average N would be ~60 or for a 30yr average N would be ~21916. As you can see for large N the formula reduces to σ_avg = σ implying that the uncertainty on monthly, annual, and 30yr averages are no different than the uncertainty on daily observations. The problem is that formula is nonsense. All texts on the propagation of uncertainty including Bevington 2003, which the author cites for this formula, clearly say that the formula is σ_avg = σ / sqrt(N). This can be confirmed via the Guide to the Expression of Uncertainty in Measurement 2008 or by using the NIST uncertainty machine which will do the general partial derivative propagation described in Bevington and the GUM for you with an accompanying monte carlo simulation. Mistake 3. The ±0.46 C figure is advertised as the annual global average temperature uncertainty. The problem is that it is a calculation only of the uncertainty for station anomaly value (annual absolute mean minus 30yr average). No where in the publication does the author propagate the station anomaly uncertainty into the gridding, infilling, and spatial averaging steps that all datasets require to compute a global average. Because the uncertainty of an average is lower than the uncertainty of the individual measurements that go into it the global average temperature uncertainty will be considerably lower than the individual Tmax and Tmin uncertainties. There are actually 3 steps in which an average is taken: a) averaging station data into a monthly or annual domain, b) averaging multiple stations into a grid cell, and c) averaging all of the cells in a grid mesh to get the global average. Not only does the author not calculate a) correctly (see mistake #2 above) but he does not even perform the propagation through steps b) and c). The point is this. That ±0.46 C figure is not the uncertainty for the global average as the author and blogosphere claim. Dr. Frank, if you stumble upon this post I would be interested in your responses to my concerns and the other concerns of those who came before me. In a future post under this thread I'll present my own type A evaluation of the monthly global average temperature uncertainty. Will it be consistent with the more rigorous analysis I mentioned above? I'll also try to periodically update the AmericanWx audience with various statistics and publications relevant to this topic. Comments (especially criticisms) are definitely welcome. I am by no means an expert in uncertainty propagation or the methods used to compute the measure the global average temperature. We can all learn together.

Per NSIDC daily sea ice extent in the Antarctic region has reached a new record low minimum value on Feb 15th, 2022 of 2.034e6 km2. This breaks the old record set on Mar 1st, 2017 of 2.080e6 km2.

Here are the updated radiative forcing equations from Etminan et al. 2016. Note that Myhre (of 5.35 * ln(C/C0) fame) is an author on this publication. The formula with minor coefficients removed for CH4 is 0.043 * (√M - √M0). For example an increase from 1900 ppb to 2500 ppb is expected to produce 0.043 * (√2500 - √1900) = +0.27 W/m2 of RF.

Relevant: Methane release from carbonate rock formations in the Siberian permafrost area during and after the 2020 heat wave

On the other side of the globe SH sea ice has quietly moved into 1st place in terms lowest extent on this date Feb. 8th according to NSIDC. We cannot eliminate the possibility of a new record low minimum down there within the next month.

0.4 MtSO2 probably isn't going to have a large effect on global average temperatures. I'm not sure we'd be able to detect it especially given that RSS and UAH have monthly uncertainties of about ±0.2 C. Of course, if scientists do determine that there was significantly more SO2 released than obviously there would be a more significant impact on global average temperatures as well.

Some elements are already on or close to land while others like the tip of the trough which will be playing a role in the cyclogenesis of the low won't hit Baja California until 18Z on Wednesday. All of this according to the NAM anyway.

I think Kuchera is overdoing it on the totals. I don't have the full Cobb output yet, but I think it's going to be about 15% less. The modeled snow depth is significantly less than what Kuchera is showing.

Interesting stat for St. Louis...we have not had a 6"+ storm (as measured at KSTL) in the month of February in 29 years! Obviously we're all rooting for this statistical fluke and futility record to finally come to an end next week.

I'm extremely suspicious of the Kuchera method here. The modeled snow depth is probably going to be more realistic at least for the STL area where ptypes are expected to be variable.

We cannot eliminate the possibility that the warming is accelerating.

All of the data is; at least all of the data I track. The 13 month centered average of the composite is now close to the trendline of +0.187 C/decade as we close out 2021.

Cheng et al. 2022 - Another Record: Ocean Warming Continues through 2021 despite La Niña Conditions Official: here Free: here

Interesting. I never thought about that until you posted. Do you think increased aerosols is a contributing factor?

Annual mean sea ice extent came in at 10.547e6 km2 for 2021. Even with a relativity great year it is still about 5% below the 1991-2020 climatological average of 11.082e6 km2. That is 9th lowest in the 42 year satellite record and if we use Walsh et al. 2016 we can extend that back 170 years. This is the first year since 2015 since annual mean sea ice extent was > 10.5e6 km2. Remember, back in 2001 the IPCC predicted that this threshold would not be crossed until about 2040. It first happened in 2007 and then again 7 other times prior to 2021. It is pretty remarkable to think that with the pattern we've seen in the Arctic this year we should have seen anomalously high extents yet we still almost dropped below this threshold again.

This the Osman et al. 2021 publication. A non-paywalled version can be found here. It's yet another hockey-stick graph. As I often tell others we have a whole hockey league of hockey-stick graphs now. They come out so often now that it's nearly impossible to keep track of them all anymore.

I thought you all might be interested in this. I created the following plot using 4 traditional, 2 satellite, 1 raob, and 1 reanalysis datasets of the global mean temperature including an equal weighted 3 month centered composite of the 8 datasets and compare everything with the CMIP5 ensemble mean from 1979/01 to 2021/08. All timeseries are using the same baseline for comparative purposes. The composite trend is +0.19 C/decade.

The NSIDC site now has an http link for their data. This is useful since many browsers no longer support FTP. https://masie_web.apps.nsidc.org/pub/DATASETS/NOAA/G02135/seaice_analysis/

Per the official 12Z NHC wind profiles the IKE is 33 TJ. Per the RAMMB profiles the IKE is the 80 TJ. I'm a little suspicious of the NHC wind profiles but if we split the difference that would give us 56 TJ. This is a pretty big deal.

Using the wind radii from the 3Z NHC advisory with VMAX = 90 kts and RMW = 10 nm the IKE has increased to 32 TJ.