Guest Post by Willis Eschenbach [note new Update at the end, and new Figs. 4-6]
In climate science, linearity is the order of the day. The global climate models are all based around the idea that in the long run, when we calculate the global temperature everything else averages out, and we’re left with the claim that the change in temperature is equal to the climate sensitivity times the change in forcing. Mathematically, this is:
∆T = lambda ∆F
where T is global average surface temperature, F is the net top-of-atmosphere (TOA) forcing (radiation imbalance), and lambda is called the “climate sensitivity”.
In other words, the idea is that the change in temperature is a linear function of the change in TOA forcing. I doubt it greatly myself, I don’t think the world is that simple, but assuming linearity makes the calculations so simple that people can’t seem to break away from it.
Now, of course people know it’s not really linear … but when I point that out, often the claim is made that it’s close enough to linear over the range of interest that we can assume linearity with little error.
So to see if the relationships really are linear, I thought I’d use the CERES satellite data to compare the surface temperature T and the TOA forcing F. Figure 1 shows that graph:
Figure 1. Land only, forcing F (TOA radiation imbalance) versus Temperature T, on a 1° x 1° grid. Colors indicate latitude. Note that there is little land from 50S to 65S. Net TOA radiation is calculated as downwelling solar less reflected solar less upwelling longwave radiation. Click graphics to enlarge.
As you can see, far from being linear, the relationship between TOA forcing and surface temperature is all over the place. At the lowest temperatures, they are inversely correlated. In the middle there’s a clear trend … but then at the highest temperatures, they decouple from each other, and there is little correlation of any kind.
The situation is somewhat simpler over the ocean, although even there we find large variations:
Figure 2. Ocean only, net forcing F (TOA radiation imbalance) versus Temperature T, on a 1° x 1° grid. Colors indicate latitude.
While the changes are not as extreme as those on land, the relationship is still far from linear. In particular, note how the top part of the data slopes further and further to the right with increasing forcing. This is a clear indication that as the temperature rises, the climate sensitivity decreases. It takes more and more energy to gain another degree of temperature, and so at the upper right the curve levels off.
At the warmest end, there is a pretty hard limit to the surface temperature of the ocean at just over 30°C. (In passing, I note that there also appears to be a pretty hard limit on the land surface air temperature, at about the same level, around 30°C. Curiously, this land temperature is achieved at annual average TOA radiation imbalances ranging from -50 W/m2 up to 50 W/m2.)
Now, what I’ve shown above are the annual average values. In addition to those, however, we are interested in lambda, the climate sensitivity which those figures don’t show. According to the IPCC, the equilibrium climate sensitivity (ECS) is somewhere in the range of 1.5 to 4.5 °C for each doubling of CO2. Now, there are a several kinds of sensitivities, among them monthly, decadal, and equilibrium climate sensitivities.
Monthly sensitivity
Monthly climate sensitivity is what happens when the TOA forcing imbalance in a given 1°x1° gridcell goes from say plus fifty W/m2 one month (adding energy), to minus fifty W/m2 the next month (losing energy). Of course this causes a corresponding difference in the temperature of the two months. The monthly climate sensitivity is how much the temperature changes for a given change in the TOA forcing.
But the land and the oceans can’t change temperature immediately. There is a lag in the process. So monthly climate sensitivity is the smallest of the three, because the temperatures haven’t had time to change. Figure 3 shows the monthly climate sensitivities based on the CERES monthly data.
Figure 3. The monthly climate sensitivity.
As you might expect, the ocean temperatures change less from a given change in forcing than do the land temperatures. This is because of the ocean’s greater thermal mass which is in play at all timescales, along with the higher specific heat of water versus soil, as well as the greater evaporation over the ocean.
Decadal Sensitivity
Decadal sensitivity, also called transient climate response (TCR), is the response we see on the scale of decades. Of course, it is larger than the monthly sensitivity. If the system could respond instantaneously to forcing changes, the decadal sensitivity would be the same as the monthly sensitivity. But because of the lag, the monthly sensitivity is smaller. Since the larger the lag, the smaller the temperature change, we can use the amount of the lag to calculate the TCR from the monthly climate response. The lag over the land averages 0.85 months, and over the ocean it is longer at 2.0 months. For the land, the TCR averages about 1.6 times the monthly climate sensitivity. The ocean adjustment for TCR is larger, of course, since the lag is longer. Ocean TCR is averages about 2.8 times the monthly ocean climate sensitivity. See the Notes below for the calculation method.
Figure 4 shows what happens when we put the lag information together with the monthly climate sensitivity. It shows, for each gridcell, the decadal climate sensitivity, or transient climate response (TCR). It is expressed in degrees C per doubling of CO2 (which is the same as degrees per a forcing increase of 3.7 W/m2). The TCR shown in Figure 4 includes the adjustment for the lag, on a gridcell-by-gridcell basis.
Figure 4. Transient climate response (TCR). This is calculated by taking the monthly climate sensitivity for each gridcell, and multiplying it by the lag factor calculated for that gridcell. [NOTE: This Figure, and the values derived from it, are now updated from the original post. The effect of the change is to reduce the estimated transient and equilibrium sensitivity. See the Update at the end of the post for details.]
Now, there are a variety of interesting things about Figure 3. One is that once the lag is taken into account, some of the difference between the climate sensitivity of the ocean and the land disappears, and some is changed. This is particularly evident in the southern hemisphere, compare Southern Africa or Australia in Figures 3 and 4.
Also, you can see, water once again rules. Once we remove the effect of the lags, the drier areas are clearly defined, and they are the places with the greatest sensitivity to changes in TOA radiative forcing. This makes sense because there is little water to evaporate, so most of the energy goes into heating the system. Wetter tropical areas, on the other hand, respond much more like the ocean, with less sensitivity to a given change in TOA forcing.
Equilibrium Sensitivity
Equilibrium sensitivity (ECS), the longest-term kind of sensitivity, is what would theoretically happen once all of the various heat reservoirs reach their equilibrium temperature. According to the study by Otto using actual observations, for the past 50 years the ECR has stayed steady at about 130% of the TCR. The study by Forster, on the other hand, showed that the 19 climate models studied gave an ECR which ranged from 110% to 240% of the TCR, with an average of 180% … go figure.
This lets us calculate global average sensitivity. If we use the model percentages to estimate the equilibrium climate sensitivity (ECS) from the TCR, that gives an ECS of from 0.14 * 1.1 to 0.14 *2.4. This implies an equilibrium climate sensitivity in the range of 0.2°C to 0.3°C per doubling of CO2, with a most likely value (per the models) of 0.25°C per doubling. If we use the 130% estimate from the Otto study, we get a very similar result, .14 * 1.3 = 0.2°C per doubling. (NOTE: these values are reduced from the original calculations. See the Update at the end of the post for details.]
This is small enough to be lost in the noise of our particularly noisy climate system.
A final comment on linearity. Remember that we started out with the following claim, that the change in temperature is equal to the change in forcing times a constant called the “climate sensitivity”. Mathematically that is
∆T = lambda ∆F
I have long held that this is a totally inadequate representation, in part because I say that lambda itself, the climate sensitivity, is not a constant. Instead, it is a function of T. However, as usual … we cannot assume linearity in any form. Figure 5 shows a scatterplot of the TCR (the decadal climate sensitivity) versus surface temperature.
Figure 5. Transient climate response versus the average annual temperature, land only. Note that the TCR only rarely goes below zero. The greatest response is in Antarctica (dark red).
Here, we see the decoupling of the temperature and the TCR at the highest temperatures. Note also how few gridcells are warmer than 30°C. As you can see, while there is clearly a drop in the TCR (sensitivity) with increasing temperature, the relationship is far from linear. And looking at the ocean data is even more curious. Figure 6 shows the same relationship as Figure 5. Note the different scales in both the X and Y directions.
Figure 6. As in Figure 5, except for the ocean instead of the land. Note the scales differ from those of Figure 5.
Gotta love the climate system, endlessly complex. The ocean shows a totally different pattern than that of the land. First, by and large the transient climate response of the global ocean is less than a tenth of a degree C per doubling of CO2 (global mean = 0.08°C/2xCO2). And contrary to my expectations, below about 20°C, there is very little sign of any drop in the TCR with temperature as we see in the land in Figure 5. And above about 25°C there is a clear and fast dropoff, with a number of areas (including the “Pacific Warm Pool”) showing negative climate responses.
I also see in passing that the 30°C limit on the temperatures observed in the open ocean occurs at the point where the TCR=0 …
What do I conclude from all of this? Well, I’m not sure what it all means. A few things are clear. My first conclusion is that the idea that the temperature is a linear function of the forcing is not supported by the observations. The relationship is far from linear, and cannot be simply approximated.
Next, the estimates of the ECS arising from this observational study range from 0.2°C to 0.5°C per doubling of CO2. This is well below the estimate of the Intergovernmental Panel on Climate Change … but then what do you expect from government work?
Finally, the decoupling of the variables at the warm end of the spectrum of gridcells is a clear sign of the active temperature regulation system at work.
Bottom line? The climate isn’t linear, never was … and succumbing to the fatal lure of assumed linearity has set the field of climate science back by decades.
Anyhow, I’ve been looking at this stuff for too long. I’m gonna post it, my eyeballs are glazing over. My best regards to everyone,
w.
NOTES
LAG CALCULATIONS
I used the Lissajous figures of the interaction between the monthly averages of the TOA forcing and the surface temperature response to determine the lag.
Figure N1. Formula for calculating the phase angle from the Lissajous figure.
This lets me calculate the phase angle between forcing and temperature. I always work in degrees, old habit. I then calculate the multiplier, which is:
Multiplier = 1/exp(phase_angle°/360°/-.159)
The derivation of this formula is given in my post here. [NOTE: per the update at the end, I’m no longer using this formula.]
To investigate the shape of the response of the surface temperature to the TOA forcing imbalance, I use what I call “scribble plots”. I use random colors, and I draw the Lissajous figures for each gridcell along a given line of latitude. For example, here are the scribble plots for the land for every ten degrees from eighty north down to the equator.
Figure N2. Scribble plots for the northern hemisphere, TOA forcing vs surface temperature.
And here are the scribble plots from 20°N to 20°S:
Figure N3. Scribble plots for the tropics, TOA forcing vs surface temperature.
As you can see, the areas near the equator have a much smaller response to a given change in forcing than do the extratropical and polar areas.
DATA AND CODE
Land temperatures from here.
CERES datafile requisition site
CERES datafile (zip, 58 MByte)
sea temperatures from here.
R code is here … you may need eyebeach, it’s not pretty.
All data in one 156 Mb file here, in R format (saved using the R instruction “save()”)
[UPDATE] Part of the beauty of writing for the web is that my errors don’t last long. From the comments, Joe Born identifies a problem:
Joe Born says:
December 19, 2013 at 5:37 am
My last question may have been a little obscure. I guess what I’m really asking is what model you’re using to obtain your multiplier.
Joe, you always ask the best questions. Upon investigation, I see that my previous analysis of the effect of the lags was incorrect.
What I did to check my previous results was what I should have done, to drive a standard lagging incremental formula with a sinusoidal forcing:
R[t] = R[t-1] + (F[t] – F[t-1]) * (1 – timefactor) + (R[t-1] – R[t-2]) * timefactor
where t is time, F is some sinusoidal forcing, R is response, timefactor = e ^ (-1/tau), and tau is the time constant.
Then I measured the actual drop in amplitude and plotted it against the phase angle of the lag. By examination, this was found to be an extremely good fit to
Amplitude as % of original = 1 – e ^ (-.189/phi)
where phi is the phase angle of the lag, from 0 to 1. (The phase angle is the lag divided by the cycle length.)
The spreadsheet showing my calculations is here.
My thanks to Joe for the identification of the error. I’ve replaced the erroneous figures, Figure 4-6. For Figs. 5 and 6 the changes were not very visible. They were a bit more visible in Figure 4, so I’ve retained the original version of Figure 4 below.
NOTE: THE FIGURE BELOW CONTAINS AN ERROR AND IS RETAINED FOR COMPARISON VALUE ONLY!! 
NOTE: THE FIGURE ABOVE CONTAINS AN ERROR AND IS RETAINED FOR COMPARISON VALUE ONLY!!
I noted in one comment that you had been ill. I hope this is all over with and nice to see you back with your heretical and artistic presentations of data. I wonder why we don’t see more of this compelling type of presentation from other quarters – did you invent this type of graph?
I noted your figures of ECS of land 0.37, NH O.17, SH 0.08 and ocean 0.03 and global 0.12. I did a weighted calc of these using ocean as 70% of globe, 87.5% of NH as land (at 0.37) and12.5% ocean and the same type of calc for SH and global and arrived at: calc global 0.13, NH 0.13, SH 0.03. Not bad. It seems that the land is reasonably homogeneous, as is the ocean. Maybe ice accounts for some of the differences.
Finally, that 31 C is so firm and appears in the darndest places. It is a constant like the freezing point and boiling point of water at sea leve. One should be able to identify this as a solid physical metric somehow – wouldn’t an equation expressing this as a law be nice? It’s in there somewhere.
tjfolkerts says:
December 19, 2013 at 11:39 am
Thanks, Tim. The data is there. Plot it as you wish. I’ve shown certain relationships. However, there are many more to explore.
w.
Your plots have made me curious about how the TOA difference appears in the output of the models (assuming they provide that). It would be an interesting validation to see if the models replicate the pattern that you’ve identified. Obviously, models that don’t are not valid representations of the earth. If they can’t match that profile then their physics is wrong. Especially if they can’t replicate the 30 deg cutoff that you show. That would seem to be a better comparison than global temperature.
Hope your recovery is proceeding well.
Gary Pearse said:
“Finally, that 31 C is so firm and appears in the darndest places. It is a constant like the freezing point and boiling point of water at sea level”.
Interestingly all three are pressure related.
http://www.animations.physics.unsw.edu.au/jw/freezing-point-depression-boiling-point-elevation.htm
just not a lot in the case of the freezing point because the volume of ice is not a lot different from the volume of water.
“because the volume occupied by a kilogram of liquid is not much different from that occupied by a kilogram of solid, this effect is very small unless the pressures are very large. For most substances, the freezing point rises, though only very slightly, with increased pressure.
Water is one of the very rare substances that expands upon freezing (which is why ice floats). Consequently, its melting temperature falls very slightly if pressure is increased. ”
Which brings us back to my contention that atmospheric pressure on the ocean surfaces determines the energy content that the oceans can hold at a given level of insolation (subject to internal ocean circulation).
http://www.newclimatemodel.com/the-setting-and-maintaining-of-earths-equilibrium-temperature/
The mass and gravity issue just won’t go away.
Temperature (heat) IS electric potential at WORK . One of the first things we learnt was to blow on our food, Why? Its all about how fast the electron moves around the molecule or solid .
Tjfolkerts, while the first graph might not be expected to be linear, it does show the relationship between Lambda and temperature pretty well and it shows lambda is inversely related to temperature, gain falls with temperature. The IPCC is therefore wrong say there is a system gain of 3, clearly effects of a doubling of CO2 get smaller and smaller, not only because the log term, but also because the gain falls, to the point that at 30 odd degrees any amount of energy causes no warming. Nobody ever talks about the doubling after this one.
I have to say this is obvious if you think about it, little stock is made from the fact that the gain, that is predominately evaporation feedbacks, are also logarithmic, the energy available for trapping/scattering, is always limited and you have to deal with law of diminishing returns as any of these gases rise. this tells me that Lamda has a log term in it somewhere, probably in the denominator
This suggest a discussion a I had with Will Kinimonth some time back has some merit. The conclusion of that was that climate change would behave like moving latitude toward the equator. Temperature will become less extreme, with lower maximums and higher minimums, with minimums rising more than maximums, New York becomes like Miami. So those of you New Yorkers who are catastrophists, but want to move to Miami, don’t bother, just wait it out.
Mr. Eschenbach:
Thanks for taking time to respond. Unfortunately, my ineptitude at quickly reverse-engineering spreadsheets has resulted in your having cast pearls before swine.
Still, I hope to try again when time permits.
Well here we go again.
I don’t know how many times we have been told that “””…CLIMATE SENSITIVITY…”””” is the increase in mean global (near) surface Temperature for a doubling of atmospheric CO2 abundance.
That after all, is the slope of the claimed logarithmic connection between CO2 and Temperature.
The solar physicists keep telling us there has been NO statistically significant change in TSI in recorded history, so how can top of atmosphere flux change ?
Could someone point us to the official SI definition of “Climate Sensitivity” Please.
On a related subject, some time back I developed a simple analog of carbon dioxide’s ability to intercept and scatter individual infrared photons as they leave the surface and pass through the troposphere, based on Nasif Nahle’s calculations of mean free photon path. Running this for a few hours gives a graphable series relating CO2 concentration to percentage of photons prevented from escaping to the stratosphere or beyond. The results show a good correlation to the theory, in that the relationship is logarithmic, with 90% of the maximum possible effect being achieved with only 30ppm CO2 at sea level pressure.
Bear in mind I’m no climate scientist, though I have studied physics. I daresay this overly simple model does not include all of the factors in the real atmosphere, and might even contain a few errors. It does clearly show the logarithmic behaviour of the greenhouse effect, though, and achieves this from first principles without any ‘adjustments’ being required.
http://iwrconsultancy.co.uk/climate/photon.png
george e. smith says:
December 19, 2013 at 3:00 pm
“The solar physicists keep telling us there has been NO statistically significant change in TSI in recorded history, so how can top of atmosphere flux change ?”
George: While the variation in radiative flux density from the sun is very small, on the order of +/-0.1% IIRC, the change in the percentage of this shortwave radiation reflected, mostly due to changes in cloud cover, and snow/ice surface cover, can be a lot larger. Similarly, the changes in longwave radiation from the earth at TOA can vary quite significantly due to temperature changes, humidity changes, and cloud cover changes. These variations are what the CERES satellites are monitoring.
george e. smith says:
December 19, 2013 at 3:00 pm
George, there are a few things at work. First, we’re talking about the solar flux that is available gridcell by gridcell … consider the gridcells right by one of the poles as example of the extreme variations in solar flux.
Next, the earth is not always the same distance from the sun, which makes an annual difference of (from memory) about 28 W/m2.
Next, the TOA flux is not just the solar radiation. It is the solar minus the upwelling longwave and the reflected shortwave. Both of these later variables change constantly.
As a result, the Net TOA imbalance varies both in space for a given time, and in time for a given location.
w.
Sensitivity is the global response willis. The assumption is that if you average the forcing globally, and average the response globally that it will be linear over the temperature of interest and time period of interest. The temperature of interest is roughly 12C to 18C. we are currently at 15C.
Steven Mosher says:
December 19, 2013 at 8:58 pm
Not sure what your point is here, Steven. Global sensitivity is the global response. Local sensitivity is the local response. Are you saying that globally averaging a wildly non-linear system somehow makes it linear?
w.
“The solar physicists keep telling us there has been NO statistically significant change in TSI in recorded history, so how can top of atmosphere flux change ?”
Well whatever they conclude about TSI the effect on surface temperature is there
http://climategrog.wordpress.com/?attachment_id=748
So rather than saying there can be no significant effect from the sun “because TSI is almost constant” someone needs look at something other than TSI, or explore what mechanism is amplifying TSI variations.
For the last 30 years they’ve been insisting it’s irrelevant, since the “pause” it suddenly becomes polite to talk about it, though they still try to ignore the fact that if it (partly) explains the “pause” it (partly) explains the late 20th. c. warming too.
Oh, dear. We won’t mention that.
Alan Robertson says: The cyclical, oscillating nature of the scribble plots reminds me of those tracing in sand by pendulums.
It’s quite analogous. Two oscillations of same period that are out of phase. In fact there is a very small difference in freq with the pendulum if the two amplitudes are different since there is a slightly non-linear relationship between freq and amplitude. That is what makes the pattern interesting as the two shift.
However, climate is not quite that simple The phase relation changes with season that’s why the shapes are not elliptic. You can get an average value of the lag from a lag-correlation plot. This would be preferable to Willis’ formula based on a clean harmonic pendulum like, oscillation.
http://climategrog.wordpress.com/?attachment_id=645
That is the usual way of estimating a phase relationship.
Willis-
Very interesting post. Thank you for your efforts.
I think the y axis labels in Figures 5 and 6 are not correct. The figure title says Degrees C per doubled CO2, but the y axis is labeled Degrees C per Watts/m^2. The figure title appears to be consistent with your text descriptions of the figures.
[Fixed, thanks. -w.]
tjfolkerts says:
December 19, 2013 at 11:39 am
If anything, you should plot the CHANGE in temperature of each grid point during the year vs the CHANGE in radiative forcing for the year.
====
Indeed. As I also posted:
Greg says:
December 19, 2013 at 5:48 am
Willis: “But the land and the oceans can’t change temperature immediately. There is a lag in the process. So monthly climate sensitivity is the smallest of the three, because the temperatures haven’t had time to change.”
No, it would be more appropriate to compare delta_d/dt(SST) to delta_Rad , the fast response is mostly orthogonal ie rate of change.
====
The fact is it’s neither one nor the other but a sliding mix of the two. http://climategrog.wordpress.com/?attachment_id=399
The monthly in-phase change will be small but the monthly (and shorter) dT/dt will be large. By the time you are looking at decades the dT/dt will be small and the response will be mainly the in-phase term.
All that is entirely consistent with a linear relaxation process , so I don’t see the bee in Willis’ bonnet being particularly satisfied with what he’s shown here.
This is very analogous to the on-going discussions of out-gassing. MacRae, Humlum etc have shown the fast orthogonal response, this is only one step to determining the long term result.
This is why there is all the talk of TCS and ECS. We are seeing some intermediary mix that is getting called TCS. I don’t think that one value is sufficient to even guess at ECS. You need a much fuller understanding of the process than one reading of a mix of the two.
“Multiplier = 1/exp(phase_angle°/360°/-.159)
The derivation of this formula is given in my post here.”
Say What?
So you are using tautochrones relationship for a diffusive process in the ground and applying it to the ocean surface well-mixed layer, with non linear feedbacks like tropical storm governor. with a whole climate system sitting on it that means the surface temp is determined by a hundred things , not of which are diffusive.
I’m in the uncomfortable position of having to agree with stoat-face on this one.
http://scienceblogs.com/stoat/
“The fatal lure of making stuff up”
Get a grip Willis. You are capable of better. This is frankly a crock. (And you don’t have the excuse of working for the government.).
Mods, if you can tell me what is that text is causing it to stick in moderation , I will do my best to avoid the repeating offence in the future.
Willis: If you haven’t realized it, you have discovered clouds (and to a lesser extent, humidity). Clouds reflect incoming SWR, which usually creates a negative radiative imbalance. Clouds also block outgoing LWR from below and radiate outgoing LWR from their top surface, so the altitude of clouds has a big influence on the radiative imbalance they create. High cold cirrus clouds can radiate so little LWR to space that they create a positive imbalance (warming). The temperature of the surface below has no impact on the radiative imbalance in cloudy areas because all of the action takes place at the top surface of the cloud. The cloudy portion of the planet therefore creates perturbs the relationship between surface temperature and TOA imbalance and the biggest perturbation occurs in the tropics where surface radiates most strongly the height of the clouds tops varies the most. (The coldest place in the atmosphere is often at the tropopause above the ITCZ – where the surface is warmest.)
At some wavelengths, humidity also blocks outgoing LWR. Actually it absorbs and re-emits it, and the temperature at the altitude from which the photons that escape to space are emitted determines the outgoing flux. For the most part humidity varies strongly with temperature, but the relative humidity is low in the downward leg of the Hadley cell and near the poles.
Fewer than 10% of the photons emitted by the surface escape directly to space, so there is no reason to expect the TOA LWR flux to correlate perfectly with surface temperature. The TOA LWR flux depends on the temperature in the atmosphere where the photons that escape to space are emitted. The more GHG’s (including water vapor) in the atmosphere, the higher the “characteristic emission level” (and temperature drops an average of 6.5 degC with every kilometer of altitude).
What matters is whether there is a linear relationship between the mean global temperature and forcing. Due to the asymmetric distribution of land, the planet as a whole warms from about 290.5 degC during winter in the NH to 294 degC in the summer. This gives us a 3.5 degC deltaT to work. The results are fairly linear. “Assessment of radiative feedback in climate models using satellite observations of annual flux variation”
http://www.pnas.org/content/early/2013/04/23/1216174110.abstract
Greg says:
December 19, 2013 at 10:49 pm
No, I’m not. Re-read the update at the end of the head post.
w.
Frank says:
December 19, 2013 at 11:05 pm
Frank, if you haven’t realized it, you haven’t discovered humility … giving me a lecture as though I’ve never heard of clouds? Get a grip, my friend.
w.
Ok Willis, I’ll check out the update. Sorry I did not see you’d changed it.
The spreadsheet you put on dropbox does not display because it seems to contain lots of refs to a local file. Could you check it is an independent file ?
“%08myfunctions1.xla’#dsin”
is that something that can be corrected by and edit or does it need some defs from another file?
Willis Eschenbach: “Then I measured the actual drop in amplitude”
That’s where my difficulty with the spreadsheet lay: the values in Sheet 2’s Q column seem to have been generated by a digitizer or something, but I don’t know where “actual” comes from.
Looking at the scribble plots , I doubt the validity of this method of assessing lag. Lag-regression seems more appropriate.
Once you have a lag, how this relates to all the different frequencies involved is complex. I suspect this is a far worse approximation than linearity. In fact I don’t see where the approximation that is being made is clearly stated.
Accepting that Willis has now dropped back to the linear relaxation equation as the base for the lag, and assuming one unique tau is enough at least to have a guess, it still comes down to a whole range of frequency dependant contributions , each at some different balance of it’s phase relationship.
http://climategrog.wordpress.com/?attachment_id=399
So is the current method here just focussing on the ‘dominant’ periodicity having the most effect on the Lissagous scribble plots?
It seems dangerous using that to estimate the decadal from the monthly. At least until it is more clearly defined why this will produce the declared result.