As usual here, I will compare the GISS and earlier TempLS plots below the jump.

Here is GISS V4

And here is the TempLS V4 LOESS-based plot

This post is part of a series that has now run for seven years. The GISS data completes the month cycle, and is compared with the TempLS result and map. GISS lists its reports here, and I post the monthly averages here.

The TempLS mesh data is reported here, and the recent history of monthly readings is here. Unadjusted GHCN is normally used, but if you click the TempLS button there, it will show data with adjusted, and also with different integration methods. There is an interactive graph using 1981-2010 base period here which you can use to show different periods, or compare with other indices. There is a general guide to TempLS here.

The reporting cycle starts with a report of the daily reanalysis index on about the 4th of the month. The next post is this, the TempLS report, usually about the 8th. Then when the GISS result comes out, usually about the 15th, I discuss it and compare with TempLS. The TempLS graph uses a spherical harmonics to the TempLS mesh residuals; the residuals are displayed more directly using a triangular grid in a better resolved WebGL plot here.

A list of earlier monthly reports of each series in date order is here:

Any chance of you reporting this using equal area graphs, such as Mollweide, rather than using the rather distorting Mercator?

ReplyDeleteYes, I promised to do so a whileago, and I have the mechanics for Mollweide and Robinson. I've been using the lat/lon plots for other things, and so hadn't made the transition. Plus, GISS only does Robinson, which I don't like much. But I'll do it.

ReplyDeleteHi Nick

DeleteWhat imho matters much more than any equal-area representation of the Globe is the question whether or not a grid output containing absolute values, anomalies or trends was subject to the same latitude weighting of the values represented by the colors in the grid as are for example time series.

If you use a Mollweide projection, only the areas will be correctly represented, but not their contents.

If we look for example at UAH's trend grid published for 1979-2021

https://www.nsstc.uah.edu/climate/2021/december2021/2021_Trend_Map.png

it is easy to see that the chart manifestly contains the same information as the simple rectangular representation anyone can generate out of UAH's 2.5 degree grid data:

https://i.postimg.cc/FRk5Cph5/UAH-2-5-degree-grid-cell-trends-Dec-1978-Dec-2021.png

The difference between UAH's latitude weighted time series for the Globe and a series generated without that weighting isn't very great, of course:

https://i.postimg.cc/Y95Qmnkq/UAH-6-0-LT-globe-orig-data-vs-Bins-grid-no-latweight.png

But taking it into account imho matters as much as the projection used to represent the areas.

And, as you noted in your explanation, Mollweide not only distorts at the Poles, but makes reading of maps unnecessarily inconvenient when you have to inspect information near the Poles.

The argument that a simple rectangular output makes Greenland appear as big as Africa is a non-sequitur which has much more to do witn ideology than with anything else.

I personally would therefore enjoy you keeping your monthly charts for us as they were all the time: easy-to-make, easy-to-read!

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By the way: could you explain us why the latitude weighting of temperatures is not simply made by using the cosine of the latitudes, but rather a much smoother weighting?

Using a formula I found years ago on the Net (for a set of latitudes, the sum of the products of the latitudinal temperature averages with the latitude cosines is divided by the sum of these cosines) gave a perfect match to Roy Spencer's monthly time series, in all four atmospheric layers.

Why is this correct?

Hi Bindidon

Delete"The argument that a simple rectangular output makes Greenland appear as big as Africa is a non-sequitur"Not entirely, I think. There is merit in the argument that the lat/lon version gives too much visual weight to the frequent great warmth in the Arctic.

But I do value the simple lat/lon viw too. I think the best thing is that I should do as I do in other situations (and as does GISS), by allowing viewers to switch between both, probably with Robinson as default, and maybe Mollweide as a third choice. Once the program is written, it is no trouble to calculate all three.

"could you explain us why the latitude weighting of temperatures is not simply made by using the cosine of the latitudes, but rather a much smoother weighting?"In calculating an average, what you really want to do is to integrate the temperature over the sphere. So if working from lat/lon grid you multiply each cell temperature by the cell's area on the sphere. In a 2.5° grid, each lat band is the same width (N to S), but while there are 144 cells in each band, the circumference diminishes as cos(lat) as you move toward the poles. Then you divide by total area, which you know anyway, but can express as the sum of all those cosines.

Thank you Nick for this quick and helpful reply.

ReplyDelete1. When I write non-sequitur, I mean: none of us can ignore that the rectangular representation of a sphere automatically exaggerates the surface of its polar regions. That is so evident that there is really no need at all to endlessly point it out everywhere.

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2. What now concerns this latitude weighting, I have to confess that I really needed your hint to definitely understand:

" Then you divide by total area, which you know anyway, but can express as the sum of all those cosines. "

Lacking your by dimensions deeper math education, this additional 'divid[ing] by total area' simply was not present in my thoughts: I saw only the weighting of the latitude bands.

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But that dividing by the sum of the cosines would matter so much in the end result I'd never imagined.

Here is a simple cosine weighting of UAH's latitude bands:

https://i.postimg.cc/5tyCXYdB/UAH-globe-orig-vs-simple-cosine-weighting.png

and here is the correct weighting

https://i.postimg.cc/VvWDRQKk/UAH-6-0-LT-globe-orig-data-vs-Bins-grid-latweight.png

As we can see, the difference between the correct weighting and no weighting at all

https://i.postimg.cc/Y95Qmnkq/UAH-6-0-LT-globe-orig-data-vs-Bins-grid-no-latweight.png

is much smaller than the difference between the correct weighting and the simple cosine weighting (which btw would give for 1979-now a trend of 0.09 °C / decade instead of currently 0.13).

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Merci beaucoup.

On the reason for equal area, I calculated the % of pixels in March 2022 which were in the dark red range, >4°C. For each of Robinson and Mollweide it was about 4%, but for simple lat/lon it was 8%. I think 4% would be correct for the sphere.

DeleteOn averaging, I think there is a simple principle that is worth being systematic about. The first step is always forming some weighted sum Σwᵢdᵢ, where d are data, w weights (could be 1). The sum might approximate an integral. Then it must be nrmalised with a factor. That should be such that the average of dᵢ=1 is 1, ie

Average=(Σwᵢdᵢ)/(Σwᵢ)

So for example the simple average of a,b,c is (a+b+c)/(1+1+1)

Seeing it this way makes a simplification that you only need the relative values of w. So you don't need the actual area of a grid cell, only that cos(lat) gives their relative value. The reason is that any constant multiplier of w in the numerator is cancelled in the denominator.

" The reason is that any constant multiplier of w in the numerator is cancelled in the denominator. "

DeleteSounds evident! But less evident is that it is a problem, because when performing latitude weighting by integrating over some latitude bands, you of course obtain the correct value, but do not know how the weighting affected the single bands themselves.

I tried to approximate Σwᵢdᵢ/Σwᵢ with an average of

(w[i]d[i]+w[i+1]d[i+1])/(w[i]+w[i+1])

over all latitude bands from the Poles to the Tropics, but the result

https://i.postimg.cc/prkkHBg3/UAH-6-0-LT-globe-orig-data-vs-Bins-grid-single-latweight.png

wasn't much better than that obtained without any weighting

https://i.postimg.cc/Y95Qmnkq/UAH-6-0-LT-globe-orig-data-vs-Bins-grid-no-latweight.png

How would you solve that problem?

Delete"How would you solve that problem?"I don't see why you should expect it to be better. It doesn't seem to approximate an integral. The reason for looking for an integral is that that says something about the function which, by construction, is independent of the way you divided up the space. Since that division is an arbitrary choice, you don't want your result to strongly depend on it. Area weighting gets its legitimacy from approximating a Riemann integral. The promise of Riemann, but really going back to Newton, is that in the limit of small subdivisions you do get an answer (the integral) which is independent of method of subdivision.

There are many ways of getting better accuracy of integral approximations.

" I don't see why you should expect it to be better. It doesn't seem to approximate an integral. "

DeleteNick, I'm not looking at anything better!

"There are many ways of getting better accuracy of integral approximations. "

I'm not looking for more accuracy of the integral; I'm simply looking at a single latitude weighting scheme.

Maybe you misunderstand why I asked for it (I wrote about that in my first comment above).

My problem is still that if e.g. UAH (and... maybe moyhu?) posts a monthly anomaly or trend grid, the data they present looks like a raw display of all grid cells containing their local anomaly or trend data, independently of their latitude.

Simply because UAH's grid output, though certainly based on a much finer resulution (see their isothermal contour splines):

https://www.nsstc.uah.edu/climate/2022/February2022/202202_Map.png

does not differ much from my raw 2.5 degree grid output:

https://drive.google.com/file/d/1YSnhfewr0v7wmkLzZXkNNV-3XQU3DlE1/view

which is based on their own data, see the files 'tltmonamg.1978_6.0' till 'tltmonamg.2022_6.0' in their directory

https://www.nsstc.uah.edu/data/msu/v6.0/tlt/

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I thought that this raw grid output cannot be quite correct.

The simple proof is that when you average the anomalies of all grid cells for a month in UAH's grid data, you obtain of course a value differing from what you compute out of the grid when using the Σwᵢdᵢ/Σwᵢ scheme (for February 2022, rounded to 2 digits atdp: +0.05 C for the raw average, instead of -0.01 C when properly integrating over all latitude bands).

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A difference of 0.06 C? That's - IMHO... - quite a lot.

And this is the reason why I was looking for a scheme allowing me to weight single latitude bands instead of integrating them into a global average.

This should be viewed as the layman's first approach:

https://drive.google.com/file/d/1AIN0zXUhRuSCUHhtHKKJu-3PpzfddZjj/view

but... the average of all cells moved only from +0.05 C down to 0.03 C instead of -0.01 C.

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Maybe you now understand better, when comparing the two grids for Feb 2022, what I'm trying to do, namely to get a grid output based on the same weighting scheme as is used to generate a time series.