A Better Way To Remove Seasonal Variations

Guest Post by Willis Eschenbach

For no other reason than my unquenchable curiosity, I took a look at the Rutgers snow cover data from KNMI. Here’s the full data as shown in the KNMI graph:

Figure 1. Rutgers snow cover extent. Note that pre-1972 there are gaps in the data.

And here’s the KNMI graph of the same data with the monthly variations removed.

Figure 2. Rutgers snow cover extent anomalies (i.e., seasonal variations removed).

When I saw that, I said “Hmmm. What’s wrong with this picture?”. Can you see what the challenge is?

(For what it’s worth, on my planet I don’t have “problems”. Instead, I have “challenges” … a small but critical difference. But I digress …)

The challenge in Figure 2 above is that there are still very large annual swings in certain places. They’re clearly visible, for example, around 1980 and can also be seen elsewhere in the record. I assume that this is because in some periods the snowfall is earlier, and in some periods it’s later. So the normal method of removing annual swings, by averaging each of the months and subtracting the monthly average of each month from the corresponding months of the raw data, simply isn’t working. It’s not properly removing the annual swings.

After pondering this for a bit, I realized that I might be able to do a better job by using a mathematical technique with the unwieldy name of Complete Ensemble Empirical Mode Decomposition. For obvious reasons, it’s usually referred to as CEEMD.

CEEMD “decomposes” any signal into a group of underlying signals which when added together reconstruct the exact original signal. It’s similar to Fourier Decomposition, but it has several advantages. I discussed the technique in my post “Noise Assisted Data Analysis“. I later wrote a post called “CEEMD and Sunspots” about how I use it frequently to see if there is an approximately 11-year cycle in climate data that would indicate if the sunspots might be affecting some given climate phenomenon.

Here is the CEEMD decomposition of the snow cover data shown above:

Figure 3. CEEMD decomposition, Rutgers snow data. The top panel shows the raw data. Panels C1 through C8 show the various empirical mode individual signals plus the residual, which when added up will reconstruct the raw data.

Clearly, the Empirical Mode C3 is the sum of all of the underlying signals that have around a one-year cycle. However, it’s not a simple regular sine wave. Instead, over time each empirical mode varies slightly in phase and amplitude. The graph below shows the raw data (blue) overlaid with the Empirical Mode C3 data (translucent red) for the early part of the record.

Figure 4. Rutgers snow data in blue, overlaid with the CEEMD Empirical Mode C3 in translucent red.

As you can see in this more detailed view above, the CEEMD Empirical Mode C3 data varies in both amplitude and phase. This is because it’s the sum of all of the underlying signals with a period around one year.

And when I subtract the CEEMD Empirical Mode C3 from the raw data, I get the following graph. I’ve repeated Figure 2 above for comparison.

Figure 5. Comparing the two methods of removing the annual cycle. Since CEEMD can only work on complete datasets without gaps, I’ve removed the pre-1972 early part of the data.

As you can see, the CEEMD method does a far better job of removing the annual cycle. It no longer contains the large annual swings shown in the standard method used by KNMI, and it clearly reveals the true underlying variations.

Why is this important? I learned early about the importance of sharp tools. My second real job, at 13 years of age for $0.30 per hour ($3.00 per hour in 2022 dollars), was digging out a foundation for a new house with a pick and a shovel. And looking back, I was probably worth about that much per hour.

In those halcyon pre-PC days, working with a shovel was called “Playing the Swedish banjo”. Here’s a recent picture of me doing that very thing:

And I’ve played the Swedish banjo for more reasonable wages a number of times since I was 13.

Perhaps as a result of my work history, I divide folks into three groups:

  • Those who have used a shovel.
  • Those who have made money with a shovel.
  • Those who have sharpened a shovel.

So I consider this new method for removing seasonal variations as sharpening a valuable tool that I use all the time. Now all I need to do is write the code to automate the process … “SMOP”, we used to call it, a “small matter of programming”.

Finally, in passing … it’s worth recalling the following prediction from 2000:

According to Dr David Viner, a senior research scientist at the climatic research unit of the University of East Anglia, within a few years winter snowfall will become “a very rare and exciting event”. “Children just aren’t going to know what snow is,” he said.

As the lower panel in Figure 5 clearly shows, that was just another one of the climate alarmists’ endless failed serial doomcasts. The mystery to me is, just why does anyone still believe them?

Anyhow, that was my day. How was yours?


PS—I’m still waiting for Twitter to work its way down to lifting my suspension. I assume they’re doing the blue-checks and the famous folks first. But if anyone who is on Twitter wanted to remind @elonmusk that I’ve been wrongly suspended, my Twitter handle is @WEschenbach. Please include a link to my post “An Open Letter To @elonmusk” discussing the crazy Twitter Rules. Many thanks.

As Usual: I ask that when you comment you quote the exact words you’re replying to. This avoids many of the misunderstandings that plague discussions on the intarwebs.

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John Hultquist
November 29, 2022 10:39 am

The CEEMD Decomposition reveals an up period prior to 1980. I was in Northern Idaho (east of Moscow/Pullman) and one of those winters produced feet of snow. I got a call from a friend at about midnight to see if I would help shovel barn roofs of a certain type; one having already collapsed.
And speaking of shovels: On trail work parties with mostly volunteers, if someone broke a shovel, they got to do the walk-of-shame at the end of the day – shovel in one hand and the blade in the other. 

John Hultquist
Reply to  John Hultquist
November 29, 2022 10:41 am

That would be:
— handle in one hand and blade in the other.

Reply to  John Hultquist
November 30, 2022 9:40 am

Yeah, when you hear the beginning of the cracking sound, you’d better let off instantly….

Reply to  beng135
December 2, 2022 11:54 am

When you hear the cracking sound it’s already too late, even if you back off. You’ve weakened the tool, it will break next time some one leans on it.

Reply to  John Hultquist
November 30, 2022 11:28 am

In 1979 it snowed so much in the Midwest that numerous indoor riding arenas collapsed. I was in North Liberty, Iowa and our arena had been built in two sections. One side of the roof eventually started to yield where the two sections were joined, so yeah, we spent some time on the roof with shovels. That was a monstrous winter and all I remember of it is snow and alcohol.

Equally memorable was a blizzard in Colorado about 15 or 16 years ago when we got 4 feet of snow out of a single storm. I had never seen that much snow from a single event. Right after that we were told children would grow up having never seen snow; and I laughed, and I laughed, and I laughed.

November 29, 2022 10:39 am

I would definitely agree with your thesis, although the variation really isn’t that great no matter how you look at it. Maybe snow volume would be better than cover but I can’t see how that would be realistically calculated to any accuracy.

The hourly rate of my first job bagging groceries was exactly the same number as one pound of ribeye steak at the time. That ratio actually held all the way to my first job after college. Unfortunately the price of ribeyes has accelerated way past the original ratio. I think now I would want to get paid in steak.

Reply to  rbabcock
November 29, 2022 2:55 pm

Greenland has comprehensive coverage of ice mass. Overall the mass is still reducing but altitude is increasing at 17mm per year:

The permeant ice extent of Greenland is also increasing. At the current rate of expansion, there will be no ice free surface by the end of the century:
comment image

And this year Greenland has a near record rate of ice accumulation:

Calving is cooling the ocean south of Greenland. It is the only region of the northern oceans that has anomalous cold. That suggests calving has accelerated. The generally warmer water around most of Greenland is melting more ice at the fringes.

November 29, 2022 10:52 am

Thanks Willis. On our family place near Round Mountain, our rototiller was four boys with shovels. Why buy gas when you’re already feeding your work force? In our dialect they were known as misery sticks.

And my first paying job was $1.25 per hour to move scorching aluminum sprinkler pipes near Redding, CA. Hot, hot, hot!

Last edited 2 months ago by McComberBoy
Pat from Kerbob
Reply to  McComberBoy
November 29, 2022 5:54 pm

75cents picking rocks in fields that grew more rocks than crop, amazing what the frost would push up every year from glacial till.

Tom Halla
November 29, 2022 10:53 am

Well, there was the “Ice ages are coming right soon now!” claims in the seventies.
I am someone who literally sharpened a shovel, several times with a file, and with a body grinder a few times.

Reply to  Tom Halla
November 29, 2022 3:08 pm

“Ice ages are coming right soon now!” claims in the seventies.

History will show that of the two climate alarms – the “ice age cometh” and”global warming”, the former is most accurate.

What is now observed as “global warming” is a sure sign that the current interglacial is coming to an abrupt end on geological timescales. The acceleration of the NH winter water cycle is impressive given the relatively subtle changes in solar intensity. The imbalance that drives snowfall only started increasing in 1400. That is just 600 years out of 9000 years and January minimum temperature on land north of 40N is increasing at 3.7C per century. That requires a substantial increase in snowfall.

November 29, 2022 11:26 am

Calling a spade a spade?

Reply to  dk_
November 29, 2022 4:27 pm

Calling a spade a bleeding shovel.

November 29, 2022 11:39 am

Hey, does anyone miss Griff, Simon or bigoilbob? Personally, I don’t miss their hijacking of discussion threads. They apparently declined to register.

Reply to  pflashgordon
November 29, 2022 1:12 pm

or their power became unreliable, or their money got diverted

Reply to  pflashgordon
November 29, 2022 2:26 pm

Oleaginous large bob was on the other day, telling lies as usual.

Pat from Kerbob
Reply to  pflashgordon
November 29, 2022 5:55 pm

B.O.B is here

Reply to  pflashgordon
November 30, 2022 9:48 am

No. They distracted but contributed nothing.

Nick Stokes
November 29, 2022 11:42 am

Yes, CEEMD is a better model, and will more effectively remove seasonal variation.

But I think it is not so much a sharpened shovel as an excavator. Monthly snow depends on month of year, but also on the amount of snow last month. So there is autocorrelation. I think an Ar(1) (ARIMA) model might be a more spade-like solution.

Mike McMillan
Reply to  Nick Stokes
November 29, 2022 12:32 pm

A few more passes thru the CEEMD filter and I’ll bet we could get a flat line.

Michael S. Kelly
Reply to  Mike McMillan
November 30, 2022 4:48 pm

A few more passes thru the CEEMD filter and I’ll bet we could get a flat line.”

I put my hard drive through Pkzip 10,000 times. One terabyte of information now occupies a single bit, whose value I wrote down somewhere before wiping the drive. I wish I could remember where I put that Post-It….

Rud Istvan
November 29, 2022 11:48 am

Nice post, WE. Seasonal detrending is a BIG deal in certain aspects of econometrics. CEEMD is not one of the several methods I was taught. Seems to work very nicely.

Rud Istvan
Reply to  Willis Eschenbach
November 29, 2022 5:32 pm

I was amazed. Cool tool, very generally useful. Some detrending examples from other domains:

  1. seasonal unemployment.
  2. seasonal income
  3. seasonal heating BTU equivalents.
Gary Pearse
Reply to  Willis Eschenbach
November 29, 2022 8:35 pm

That sharp ridged zigzag curve looks built to to persist forever! I’m betting that snow-a-thing-of- the-past Viner retired to a Greek Isle and doesn’t watch the news.

Re the shovel, I’ve dug into a variety of media. Had a mixed farm with dairy cow (and calves), sheep, horse, pigs, chickens, ducks and geese, veggies, oats and corn and lots of winter snow in Eastern Ontario, Canada.

I also dug a basement in Leysin, Switzerland on my odyssey in the early 1960s to top up meager savings. For a prairie boy, it was a bit different, being on a fairly steep mountain slope. The back wall, after digging, was about 9 feet and there were a few boulders to be dug out and carefully maneuvered and embedded in the growing terrace of earth out front. I was working with another guy.

Sure enough, a 300 pounder got loosened by a little overnight rain and the next day, back at work, this boulder took off! I thought it would be arrested in a patch of forest below, but it ripped through and we heard the most mighty crash in the distance! We made our way down slope to see what had happened and arrived at the top of a road cut to see it had flown over the road and into large pile of lumber. We decided to take the rest of the day off.

Clyde Spencer
November 29, 2022 12:36 pm

In my real job before retiring, I was often confronted with the task of improving satellite imagery for purposes of interpretation or intelligence gathering.

An approach that I found useful was in recognizing that there were different kinds of corruption in the imagery, and they required different techniques.

A typical schema was to start with several bands of a mutispectral image, such as Landsat. I’d do a Principal Components (PC) decomposition to start. Usually, random noise would be found in the highest order Principle Components. A 3X3 Median Filter was effective in reducing the Shot Noise and smooth things out. Periodic noise, commonly 60-cycle hum, would show up in one or more intermediate bands. I’d do a Fourier Transform of the PC and suppress the periodic noise. I might also run an edge-sharpening filter over the first PC, because it invariably carried all the intensity information of the image. I would then reverse the PC operation to derive the original bands used, with the changes. Then, I could select three bands to compose a working image. The results were usually dramatic. The human eye is very good at discerning the quality and utility of an image.

In this approach, the periodic influence would be akin to your C3 component. My approach can be applied to data of lower dimensionality such as a a time series. I’d suggest fitting a linear (or if apparently justified, a higher order) regression to the C3 component before recovering the original data.

November 29, 2022 1:04 pm

“crazy Twitter rules” could also be called criminal Twitter rules of targeted censorship of those with a beautiful mind.

November 29, 2022 1:11 pm

Now can we do cycle comparisons with La Nina in turning off the water supply? and grouped solar cycles for slow recharge over extended periods? and the AMO for longer departures from trend?

November 29, 2022 1:22 pm

All of this snow analysis fails to identify the key fact with snow at the present stage of the intensifying of the winter water cycle. The maximum extent is trending upward at 56,000km^2 each year.

At the present time, the only identified region increasing permanent ice cover is Greenland. It will be 100% permanent cover by the end of this century.

The trend for snowfall at Mt Rainier has been upward for the last 100 years – per attached Paradise data.

The energy imbalance that will end the current interglacial started in J1400. Some time after that the northern oceans began to warm and the winter water cycle over the NH has been ramping up for at least 100 years. Only Greenland is accumulating ice extent and altitude now but calving still outpacing the gain in total ice mass.

The January temperature on land north of 40N has increased by 3 degrees in the last 70 years. The only way that can happen is increased advection from the oceans resulting in higher snowfall.

The July temperature of the oceans north of 20N is increasing at 2.9C per century. So surface and near surface heat content rising rapidly.

Oceans store heat and land stores ice. So far melting is still out pacing snowfall but the NH is only 600 years into the first 9000 years of the initial glaciation. Record snowfall will be a feature of the next 9000 years.

Reply to  RickWill
November 29, 2022 1:38 pm

The last two glacial cycles persisted for 4 precession cycles. So the next 9000 years will only be the first phase. The intensity of the Gulf Stream increased once the Panama Isthmus formed and that increased the depth of glaciation. Glaciation is ended once the glaciers are calving at sufficient rate to shut down the winter water cycle. Once the sea level starts rising, calving accelerates and the surface temperature of the oceans collapse.

A key question for any budding climate scientist is why should the termination of the current interglacial be any different to the last four. All interglacials. ended with the summer solar intensity over the NH ramping up.

The “global warming” that ended the last three interglacials would have been more intense than the present era. The earliest termination on the attached chart at -399k would have similar “global warming” to the present time. But sea level still fell rapidly.

The CO2 demonisers, with their special brand of phiisics, are mislabelling a truly historic event for human civilisation. It is no wonder these incompetent clowns are baffled.

Hands up if you knew Greenland had gained 170mm in elevation, on average, over the past decade.

Reply to  Willis Eschenbach
November 29, 2022 2:13 pm

Not seeing it.

Why would you look at Antarctic temperature for anything to do with interglacial? Antarctica is permanent ice block. It has not had an interglacial.

The best way to identify the end of an interglacial is falling sea level as I have shown in the attached chart above – top chart in the series – open in new tab to get full size. You know then that the northern land masses, that are now ice free, start to gain ice again.

Last edited 2 months ago by RickWill
Reply to  Willis Eschenbach
November 29, 2022 3:47 pm

With regard point 1). It is an industry that believe CO2 somehow changes Earth’s energy balance, there is back radiation from cold objects to warm objects and glaciation is associated with low heat input. So no hope for that industry. Muddled phiisics of their own design dominates that industry.

For the Sea Level Reconstruction, I used the NOAA/Spratt data set:

Sea level is by far the most direct indicator of glaciation.

If you did a Fourier analysis of the Antarctica temperature data you should find a strong component at 23kyr, Indicative of the precession cycle but there is a lot of noise in the temperature signal in Antarctica related to glaciation.

Factors creating noise in the Antarctic temperature are:

  1. The warming and cooling cycle of the northern and southern oceans are mostly out of phase.
  2. The land temperature over glaciated land is much colder than unglaciated land for two reasons – lapse rate due to average surface increase of 600m over the sea level and ice blocks are hard to get warmer than 0C.
  3. Eventual ocean cooling due to glacier calving takes a long time.

So the ways these factors interplay with glaciation in the NH is far more complex than the sea level.

The sea level reconstruction has a dominant peak at 23.5kyr consistent with precession driving it. I would be surprised if precession is not identifiable in the Antarctic temperature.

Reply to  RickWill
November 29, 2022 7:00 pm

You seem to have a lot of insight into this. How about presenting an article with all the data analysis supporting your hypotheses so we can better understand it?

Last edited 2 months ago by stinkerp
Reply to  stinkerp
November 29, 2022 7:38 pm

My thoughts exactly. I want to learn to this to the bone.

Reply to  stinkerp
November 29, 2022 8:18 pm

 How about presenting an article with all the data analysis supporting your hypotheses

Calling it a hypothesis is a stretch. I simply accept that climate change is a constant; is not something that only occurred since 1850 and is not something driven by a change in concentration of a non-condensing trace gas in the atmosphere.

I am working on collating the data on the role precession plays in the water cycle and glaciation as it has done four times in the last 500kyrs. One basic understanding is that liberating water from the oceans to deposit on land as snow is highly energy intensive. It requires “global warming” (at least a lot more heat in the northern oceans) to get going.

I have sorted how snow melts (that might be obvious but snow would not melt anywhere on earth ON AVERAGE – the average solar absorbed by snow would not overcome long wave losses). Determining the rate of snowfall is proving harder than I expected (there is much higher proportion of sensible heat advection than I anticipated so it cannot be neglected in the snowfall calculations).

My goal is to predict the southward advance of the northern permafrost once snowfall overtakes snow melt. I understand the permafrost line is still advancing northward. Greenland has turned the corner with almost 100% permanent ice cover now and will be by the end of the century.

Reply to  RickWill
November 29, 2022 3:15 pm

The July temperature of the oceans north of 20N is increasing at 2.9C per century. So surface and near surface heat content rising rapidly.”

What source measures ocean temperatures north of 20N?
Then, what length of time are you basing this claim?

Reply to  ATheoK
November 29, 2022 4:02 pm

The best dataset for ocean surface temperature is NOAA/Reynolds OI. It uses fixed and moving buoys for temperature refenence and satellite data for interpolation. It is readily available on climate explorer:
First two years for 20 to 90N all oceans and land locked water bodies:
1981 -999.9000   -999.9000   -999.9000   -999.9000   -999.9000   -999.9000   -999.9000   -999.9000   -999.9000   -999.9000    14.33930    12.85688   
 1982  11.81331    11.31182    11.29090    11.72291    12.98239    14.75637    16.61943    17.77998    17.47606    15.96479    14.28680    12.83685 
Last two years:
 2021  12.42951    11.84068    11.80335    12.30309    13.57746    15.50746    17.64307    18.91117    18.60937    17.00613    15.30472    13.53506   
 2022  12.33269    11.75278    11.71819    12.40009    13.68928    15.74784    18.02373    19.27096    18.95615    17.30568   -999.9000   -999.9000  

You can get the full set from Climate Explore with 1×1 degree resolution.

Reply to  Willis Eschenbach
November 29, 2022 6:01 pm

You have the whole year. Take a look at just the July anomaly.

My point is that it is rising rapidly in July. It is driven by the dramatic reduction in advection of heat to land during the northern summers when the summers land temperature are high.

The reduction in summer advection, leaves more heat in the oceans that is available to enhance the winter water cycle when the land sunlight is lowest.

Last edited 2 months ago by RickWill
November 29, 2022 2:30 pm

So we have a more concentrated variation than the 1970s with it’s big highs and big lows. I am looking forward to future big highs in snow cover because I love snow.

Pat from Kerbob
Reply to  JC
November 29, 2022 6:00 pm

Please move to calgary as starting jan1 I start running out of snow goodwill.

All yours

November 29, 2022 2:51 pm

Unfortunately Willis you’ve fallen victim to lack of context for Dr Viner’s quote. He was specifically referring to the SE of England which has seen diminishing amounts of snow over the last few decades. He even added the following, which hardly ever gets quoted: Heavy snow will return occasionally, but when it does we will be unprepared. We’re really going to get caught out. Snow will probably cause chaos in 20 years time,”. Which is exactly what happened ten years later in London. Where I grew up the local authorities’ snow ploughs were no longer getting used regularly so they stopped paying for them so the rare heavy snow fall caused chaos.

Reply to  Willis Eschenbach
November 29, 2022 9:57 pm

“Unfortunately, Phil, you’re defending the indefensible.
You are seriously claiming that Viner was correct in saying that snow in SE England would become “a very rare and exciting event”, that “Children [in SE England] just aren’t going to know what snow is”, and that “Snow [in SE England] will probably cause chaos in 20 years time“.
Those not only didn’t come true, they didn’t come close to coming true.”

I was saying that Viner’s quote is frequently taken out of context, which is what you did too. Unlike most who I’ve criticized for doing so you have at least accepted the point and tried to address it. However Viner made his comment in 2000, all the data you have presented is ‘the future’ from his perspective.
Here’s some data running back to 1947 (the year my dad always referred to when I was growing up as a standard by which all winters should be judged; until 62/63):
comment image
As you’ll see the decade prior to 2000 showed a series of winters with negligible snowfall. The median for snow-lying days during the period of the data is 6days over the last 30 years it’s 2 so the trend is in the direction Viner predicted. He was certainly correct about the occasional heavy snow storm causing chaos now, an inch of snow in London will bring the city to its knees.

Reply to  Willis Eschenbach
November 30, 2022 9:08 pm

“What part of “the trend since Viner’s prediction is a slight INCREASE in snowfall” seems unclear to you? Look at the red trend line in the graph just above your comment.”

Actually it shows no increase/decrease.

“Viner predicted a DECREASE in snowfall, and not a small decrease—a decrease so extensive that kids wouldn’t know what snow is”. 

And if you read the article in which he is quoted you’d see they were discussing what had happened over the previous decade, which was a period of low snow coverage and what the implications for the future were.

The data which I referred to from 1947 to present covers that period. The 30yr median from 1947 is 8.5 days/yr, from 1967 it’s 5 days/yr, from 1987 it’s 2 days/yr. Over that whole period there were 11 years when there were zero days of snow coverage, 10 of which were since 1990 as opposed to once in the first 40 years of the period.

Reply to  Phil.
December 1, 2022 5:44 am

Your response above even notes that there “isn’t a decrease”. Even were that true, it would make Viner wrong.
Viner was wrong, plain and simple, because he exaggerated or misread or misinterpreted trends.

Reply to  c1ue
December 1, 2022 9:42 am

No, there is no significant change in the data you showed which only covered the last 20 years and was very erratic, I would think if you’d calculated the t-test you would have found the p-value to be statistically insignificant.
The comparable data I produced from 1947 shows that the 30year median has decreased by a factor of 4 over that period and that the probability of zero snow cover in a given year is much higher recently, so the occurrence of snow fall in SE England is decreasing.

Reply to  Willis Eschenbach
December 2, 2022 9:27 am

Well I tried to lead you to a more complete set of data which illustrated what Viner was talking about but you refused to drink(sic). At least now you know the context of his remarks and presumably won’t misrepresent it in future.

Reply to  Phil.
November 29, 2022 7:02 pm

Who is the Mayor of London again? Could that have anything to do with London not being ready for,(fill on the blank)?

Reply to  Drake
November 30, 2022 9:09 pm


November 29, 2022 4:03 pm

I think you will find that Viner’s quote was specifically talking about Britain. As is clear when you see where the quote can from:

While still derisible, constantly taking it out of context by applying it to things like the total Northern Hemisphere does us no favours.

michael hart
November 29, 2022 4:13 pm

“Perhaps as a result of my work history, I divide folks into three groups:

Those who have used a shovel.
Those who have made money with a shovel.
Those who have sharpened a shovel.”

Actually, the veins on your hand make you look like a rock climber. I used to do it a bit.

Loren Wilson
Reply to  michael hart
November 29, 2022 5:30 pm

According to Clint Eastwood in “The Good, the Bad and the Ugly”, there are only two kinds of men – those with shovels and those with guns. He had the gun and Tuco dug.

Izaak Walton
November 29, 2022 5:38 pm

You have an interesting definition of “better”. You state that
“As you can see in this more detailed view above, the CEEMD Empirical Mode C3 data varies in both amplitude and phase.”
which means that you are removing some fraction of both the annual variation and also the information about when it was snowing.

For example you say that “in Figure 2 above is that there are still very large annual swings in certain places. They’re clearly visible, for example, around 1980 and can also be seen elsewhere in the record. I assume that this is because in some periods the snowfall is earlier, and in some periods it’s later.”

now the phase information tells you about when it snowed while the amplitude tells you about how much snow there was in a particular year. Subtracting off a signal that varies in both amplitude and phase means that you no longer have that information.

Now the basic point is that there is no universally agreed best way to subtract off the annual variation. Rather there is an optimal one for each particular analysis. So if I were to try and look at whether the snowfall was early or late in each year your method would fail. Similarly if I wanted to know about the total amount of snow each year.

Izaak Walton
Reply to  Willis Eschenbach
November 29, 2022 9:08 pm

If you remove a periodic function corresponding to the average annual snowfall then what removes in not only the trend but also the year to year flucuations. If on the otherhand you remove a CEEMD mode then you are removing not only the periodic function but also some of the year to year flucuations. If you want to look at long term trends then that makes sense but if you want to look at changes from year to year then you are losing important information.

I am guessing but suppose you had a signal represented by a(t)*sin(w t) + noise
where a(t) is slowly varying compared to the frequency w. Then the CEEMD decomposition would have one mode looking like a(t)*sin(w t) and if you subtracted that off you would lose all the information about the long term trend.

Pat from Kerbob
November 29, 2022 5:52 pm

Certainly can’t argue with the original graph for south Saskatchewan.

The 70s were megasnow, blizzards where we could drive skidoos on top of the school, and jump off the gym into snow banks, 35’ high roof.
Then in winter of 80-81 basically no snow, like flipping a switch, the beginning of drought years. Years where farmers would get less than 1“ rain.

Reply to  Pat from Kerbob
November 29, 2022 6:30 pm

I wonder if the low of 80-81 was the result of Mount St Helens erupting?

The January temperature on land north of 40N rose strongly from 1948, the beginning of the GHCN data, till 1963 then dropped suddenly before continuing to trend up again. I put that shift down to the Alaskan landslide upsetting the regional climate.

St Helens erupted in March 1980. The temperature anomaly for land north of 40N in January 1981 was 3.5C warmer than January 1980.

November 29, 2022 6:33 pm

Can’t we just tilt the Earth up a bit so that its axis is perpendicular to the orbital plane? Surely that would remove all seasonal variations.

Christopher Chantrill
November 29, 2022 9:13 pm

But Willis! Here in North Seattle it just snowed, on November 29: the earliest I can remember.

What does it mean? What do the climate priests say? Will everything be all right?

November 30, 2022 8:42 am

Willis, have you looked at Winter’s method:

It’s heuristic, but tunable. There’s nothing wrong with tuning models as long as you respect its limitations. One way of tuning is to use smoothing constants, each of which is stepped over a given range. The model is then run with each value and the results are compared to the data and optimized by minimizing the sum of squared errors.

After this computationally intensive exercise you end up with an estimate of the uncertainty of the model (i.e. the minimized sum of squared errors), although I have never seen such an estimate published for climate models.

That estimate of uncertainty can be likened to an instrument calibration and, therefore, is only valid within the calibration range. I would think the minimized mean squared error would be a valid estimate of the uncertainty of the forecast for the first forecasting period, although I think the forecasting error would probably grow exponentially for subsequent forecasting periods. Just a thought.

Forecasts for tuned models can be compared to making measurements with a calibrated instrument, but outside the calibration range: the further out of the calibration range, the greater the uncertainty of the forecast (or measurement, in the case of an instrument).

Last edited 2 months ago by Phil
November 30, 2022 9:24 am

But, WHY? Why torture an already tortured data set to tell us something (what?) that one might think is not in the original data?

The original set above is “Snow Cover Extent, averaging anomalies over region [northern hemisphere]…monthly mean of daily NOAA/NCEI Climate Data Record of snow cover extent”.

That is a “data set” but why look at it, why fuss with it further? It is already hopelessly compromised by more than one “averaging” step. Anomalies themselves are departures from some average, then they average the anomalies — can anyone say “What the heck?”

w. calls all that “… the normal method of removing annual swings, by averaging each of the months and subtracting the monthly average of each month from the corresponding months of the raw data, simply isn’t working.” I am not surprised it isn’t working….it isn’t worth doing — it obscures the information the data set was collected to reveal.

Comparing two over-manipulated data sets is interesting the the data handling world, but here they are, overlaid:

One sees the difference, the loss of detail, the CEEMD versions losses the extremes, changing the sign in many cases.

Last edited 2 months ago by Kip Hansen
Frank from NoVA
Reply to  Kip Hansen
November 30, 2022 2:21 pm

Exactly right, Kip. I like ‘modeling’ as much as the next person, but the cardinal rule is that graphing the data should always be the first step of any analysis before unleashing the heavy artillery. In this case, if you were to tell me that the ‘data’ in Figure 1 was accurately represented, my first response would be that no other processing is required since there clearly isn’t anything going on with snow extent.

Forrest Gardener
Reply to  Kip Hansen
November 30, 2022 4:37 pm

Thank goodness somebody with your credibility asked the question of why?

I was waiting for an inference and it never came. And I was left with an analogy in my own mind that after you pulp a forest you end up with a uniform pile of woodchips just ready to be made into paper.

Frank from NoVA
Reply to  Willis Eschenbach
November 30, 2022 5:21 pm

‘For the same reason that the government reports the seasonally adjusted unemployment figures.’

Willis, many government ‘seasonable adjustments’ (sa). particularly with respect to economic data, are notorious for turning ‘bad’ news into ‘good’ news and vice versa, so I would advise some caution in modeling the sa data. Better to use the raw data, which is usually available.

Frank from NoVA
Reply to  Willis Eschenbach
December 1, 2022 9:11 am


Here’s a link to the BLS tables for CPI adjustment:


I only opened up the first table (2017-2021), the first several rows of which were sufficient to show that BLS varies their monthly adjustment factors each year.

As noted above, I specifically made reference to ‘economic data’. While I’m sure our government’s economic statisticians are highly qualified, they make the mistake of believing that the economic actions of human beings are measurable in the way of physical processes.

If you’re interested, here’s a link that explains why seasonable adjustments to economic data are problematic:

Forrest Gardener
Reply to  Willis Eschenbach
November 30, 2022 6:26 pm

Thanks Willis. So what was your inference?

Forrest Gardener
Reply to  Willis Eschenbach
December 1, 2022 11:50 pm

Let’s stick with the head post. You say you described a couple of ways to remove seasonal variations.

Having removed those seasonal variations, did you make any inferences about anything?

Forrest Gardener
Reply to  Willis Eschenbach
December 3, 2022 3:00 pm

Thanks. Sorry to be so dim.

November 30, 2022 1:15 pm

Interesting analysis Willis. My PhD in Econometrics has been gathering dust for a long time. Even so, many of the climate questions revolve around issues of stationarity. That is, do climate variables of interest have constant means? There are tests to answer this question. For example, I did a quick excel calculation of the Dicky Fuller test for a unit root of the NH snow coverage data. You can not reject a constant mean from the data. THe problem with most measured climate data is that we have very short measured series on variables with potentially very long and complicated cycles. It seems evident from the ice core data that climate (temperature) has a bi-stable distribution over the long term.

It seems that seasonal differencing is the right approach if you want to examine an ARIMA model (autoregressive integrated moving average.) as Nick suggested. ARIMA models are simple but powerful tools for building forecast models. I have always been befuddled about why climate science uses anomalies calculated as differences from an average value calculated over some subperiod rather than just seasonal differencing.

While I haven’t checked all of the CRN stations, few have positive and significant time trends over the period measured. The starting point is taking seasonal differencing. I wonder if your approach to seasonal adjustment would yield different results. Last I checked, none of the seven CRN stations in CA had significant positive temperature trends.

Reply to  Willis Eschenbach
December 1, 2022 7:38 am

And how did it compare to the method you are referencing in this post, the CEEMD decomposition? Apparently Mr. Stokes got your attention….

November 30, 2022 1:56 pm

Figure 2. Rutgers snow cover extent anomalies (i.e., seasonal variations removed).”

No, showing monthly anomaly is not removing seasonal variations. The extent dec-jan 1979 and 1980 really was low.

December 2, 2022 11:51 am

I agree that Willis’ processing has slightly less residuals and some interannual variations in 1980s are more visible. However, unless you want to preserve the subannual noise, the best thing is a simple low pass filter.

It would be interesting to see the Rutgers snow data with a triple running mean but I’m away from base at the moment and don’t have by processing tools to hand.

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