This post is an update on my earlier post on the same subject.  In my earlier post I regressed linearly detrended Hadley global temperature anomalies since 1950 against a combination of low pass filtered volcanic forcing, solar forcing, and ENSO influence.  For ENSO influence, I defined a low pass filtered function of the detrended Nino 3.4 index, which I called the Effective Nino Index (ENI).

My regression results suggested the rate of warming since 1997 has slowed considerably compared to the 1979 to 1996 period, contrary to the results for Foster & Rahmstorf (2011), which showed no change in the rate of warming since 1979.   There were several constructive (and some non-constructive) critiques of my post in comments made here at The Blackboard…. and elsewhere.  This post will address some of those critiques, and will examine in some detail how the choices made by F&R generated “no change in linear warming rate” results; results which are in fact not those best supported by the data.

Limitations of the Regression Models

 It is important to understand what a regression, like done by F&R or in my earlier post, can and can’t do.  If each of the coefficients which come from the regression are physically reasonable/plausible,  then the quality of the regression fit indicates:

1) if the assumed form of the underlying secular trend is plausible, and

2) if there are important controlling variables not included in the regression

The difference between the regression model and the data (that is, the model ‘residuals’) is an indication of how well the regression results describe reality: Is the form of the assumed secular trend plausible? Do the variables included in the regression plausibly control what happens in the real world?   However, if the coefficients from the regression output are not physically reasonable/plausible, then even a very good “fit” of the model to the data does not confirm that the shape of the assumed secular trend matches reality, and the regression results may have little or no meaning.

Some Substantive Critiques from My Last Post

The fundamental problem with trying to quantify and remove the influences of solar cycle, volcanic eruptions, and ENSO from the temperature record was pointed out by Paul_K, who noted that selection of a detrending function, which represents the influence of an unknown secular trend, is essentially circular logic.  The analyst assumes the form of the chosen secular function (how the secular function varies with time: linear, quadratic, sinusoidal, exponential, cubic, etc, or some combination) in fact represents the ‘true’ form of the secular trend.  The regression done using a pre-selected secular function form is then nothing more than finding the best combination of weightings of variables in the regression model which will confirm the form of the assumed secular trend is correct.

Hence, any conclusion that the regression results have “verified” the true form of an underlying trend is a bit circular… you can’t verify the shape of an underlying trend, you can only use the regression to evaluate if what you have assumed is a reasonable proxy for the true form of the secular trend.  In the case of F&R, the assumed shape of the secular trend was linear from 1979; in my post the assumed secular trend was linear from 1950.  Both suffer from the same circular logic.   F&R also allow both lag and sensitivity to radiative forcing to vary independently, which allowed their regression to specify non-physical lags and potentially non-physical responses to forcings, which together lead to the near perfect ‘confirmation’ of their assumed linear trend.   All of the regressions in this post, as well as in my original post, require that both solar and volcanic forcings to use the same lag, though that lag is free to assume whatever value gives the best regression fit, even if the resulting lag appears physically implausible.

Nick Stokes suggested substituting a quadratic function (with the quadratic function parameters determined by the regression itself) and went on himself to compare the regression results for linear and quadratic functions for 1950 to 2012 and 1979 to 2012.   Like me, Nick used a single lag for both solar and volcanic influences.  Nick concluded that with a quadratic secular function, there is some (not a lot) deviation from a linear trend post 1979, which varies depending on what temperature record is used.  Nick’s results are doubtful because simply choosing a quadratic secular function is just as circular as choosing a linear function.   Some of the lag constants Nick’s regression found for the 1975 to 2012 period (eg. ~0.11) appear physically implausible (much too fast).

Tamino (AKA Grant Foster of F&R) made a constructive comment at his blog: a single lag constant for solar and volcanic influences (a “one box lag model”) was not the best representation of how the Earth is expected to react to rapid changes in forcing like those due to volcanoes, and that a two-box lag model with a much faster response to account for rapid warming of the land and atmosphere would be more realistic.  I have included this suggestion in my regressions.

Commenter Sky claimed that basing ENI on tropical temperature responses was “a foolishness” (I strongly disagreed) but his comments prompted me to look for any significant correlation between the ENI and non-tropical temperatures at different lags, and I found that there is a very modest but statistically significant influence of the ENI of non-tropical temperatures at 7 months lag.  Incorporating both ENI and 7-month lagged ENI slightly improves the regression fit in all cases I looked at, and generates an estimated global response for ENI (not lagged 7 months) which is close to the expected value of half the response for the tropics.   (I will describe the (modest) modifications I made based on Tamino’s suggestion and on a 7-month lagged ENI contribution in a postscript to this post.)

Finally, Paul_K suggested that a way to avoid logical circularity was to try a series of polynomials in time, of increasing order, to describe the secular trend (with time=0 at the start of the regression model period) and with the polynomial constants determined by the regression itself.  The resulting regression fits can then be comparing using a rational method like AIC (Akaike Information Criterion) to determine the best choice for order for the polynomial (the minimum AIC value is the most parsimonious/best).  For a linear regression with n data points and M independent parameters, the AIC is given approximately by:

AIC = n * Ln(RSS) + 2*M

Where Ln is the natural log function and RSS is the Residual Sum of Squares from the regression (sometimes also called the “Error Sum of Squares”).  M includes each of the variables: solar, volcanic, ENI, Lagged ENI, secular function constants, and the constant (or offset) value.  Higher order polynomials should allow a better/more accurate description of secular trends of any nonlinear shape, but each added power in the polynomial increases the value of M, so a better fit (reduced RSS) is ‘penalized’ by and increase in M.   A modified AIC function, which accounts for a limited number of data points (called the AICc) is better when the ratio of n/M is less than ~40, but this ratio was always >40 for the regressions done here.

AIC Scores for Polynomials of Different Order

The ‘best’ polynomial to use to describe the secular trend, based on the AIC, depends as well on whether or not you believe that the influences of volcanic forcing and solar forcing are fundamentally different on a watt/M^2 basis.  That is, if you believe that solar and volcanic forcings are ‘fungible’, then those forcings can be combined and the regression run on the combined forcing rather than the individual forcings.   In this case, the best fit post 1975 is quadratic.  Troy Masters (Troy’s Scratchpad blog, based on a suggestion from a commenter called Kevin C) has showed that summing the two forcings improves a regression model’s ability to detect a known change in the slope of secular warming in synthetic data.

If a regression is done starting in 1950 (as in my original post) with solar and volcanic forcings treated separately, then it appears the best, or at least most plausible polynomial secular trend is 4th order, which represents a ‘local minimum’ in AIC… lower than 5th order.  AIC scores of 6th order and above are smaller than 4th order,  but the regression constants for solar and volcanic influences do not “converge” on similar values for each higher order polynomial; they instead begin to oscillate, indicating that the higher order terms (which can simulate higher frequency variations) are beginning to ‘fit’ the influences of volcanoes and solar cycle, rather than a secular trend.  In any case, using  a 4th order polynomial for the regression starting in 1950 generates a much improved fit compared to an assumed linear secular trend.

1975 to 2012

Figure 1 shows the AIC scores and lag constants for regressions from 1975 to 2012.

 

The minimum AIC score is for a third order polynomial.  The corresponding regression coefficients (with +/- 2-sigma uncertainties) are shown below:

 

Volcanic                                0.14551 +/-0.0248

Solar Cycle                           0.47572 +/-0.2034

ENI                                         0.08253 +/-0.0143

7 Mo Lagged ENI                 0.03179 +/-0.0144

Linear Contribution             0.01335 +/-0.00897

Quadratic Contribution      0.0003753 +/- 0.000542

Cubic Contribution              (-8.655 +/-9.18)*10^(-6)

Constant                                0.02384 +/-0.0383

R^2                                         0.828

F Statistic                               306

Figure 2 shows the regression model overlaid with the secular component of the model.  The secular component is what is described by the above linear, quadratic, cubic contributions plus the regression constant.  Figure 3 shows the regression model overlaid with Hadley temperature anomalies.   The model matches the data quite well.  The residuals (Hadley minus model) are shown in Figure 4.  The residuals are reasonably uniform around zero, and show no obvious trends over time.

Figure 5 shows the individual contributions (ENSO, solar cycle, volcanic) along with their total.

Figure 6 shows the original and ‘adjusted’ Hadley data, where the influences for ENSO, solar cycle, and volcanoes have been subtracted form the original Hadley data.  I have included calculated slopes for 1979 to 1997 (inclusive) and 1998 to 2012 (inclusive).  The best (most probable) estimated trend for 1979 to 1997 is 0.0160 C/yr, while from 1998 to 2012 the best estimate for the trend is 0.0104C/yr, corresponding to a modest (35%) reduction in the rate of recent warming.  (edit: 0.0160 should have been 0.0171, 0.014 should have been 0.0108; the reduction is 37%)

 

1950 to 2012

Figure 7 shows the AIC scores and lag constants for regressions from 1970 to 2012.

The local minimum (best) AIC score is for a fourth order polynomial.   At orders 6 and above the AIC score continues to fall, but without convergence of solar and volcanic coefficients, which suggests to me that the higher order polynomials are beginning to interact excessively with the (higher frequency) non-secular variables we are trying to model, and the continued fall in AIC score is not indicative of a true improvement in accuracy of the higher order polynomials as a secular trend.  I adopted the fourth order polynomial as the most credible representation of the secular trend.  The corresponding regression coefficients (with +/- 2-sigma uncertainties) are shown below:

Volcanic                                0.1346 +/-0.0234

Solar Cycle                           0.3283 +/-0.170

ENI                                         0.0972 +/-0.0122

7 Mo Lagged ENI                 0.0245 +/-0.0121

Linear Contribution             0.00804 +/-0.00828

Quadratic Contribution       -0.000772 +/- 0.000542

Cubic Contribution              (2.97 +/-1.32)*10^(-5)

Quartic Contribution           (-2.7 +/-1.06)*10^(-7)

Constant                                -0.0137+/-0.0374

R^2                                         0.83691

F Statistic                               473

Figure 8 shows the regression model overlaid with the secular component of the model.  The secular component is what is described by the above linear, quadratic, cubic, and quartic contributions plus the regression constant.  Figure 9 shows the regression model overlaid with Hadley temperature anomalies.   The model matches the data quite well.  The residuals (Hadley minus model) are shown in Figure 10.

Figure 11 shows the original and adjusted Hadley data, where the influences for ENSO, solar cycle, and volcanoes have been subtracted form the original Hadley data.  I have included calculated slopes for 1979 to 1997 (inclusive) and 1998 to 2012 (inclusive).  The best estimate for the trend from 1979 to 1997 is 0.0164 C/yr, while from 1998 to 2012 the best estimate for the trend is 0.0084C/yr, corresponding to a 49% reduction in recent warming.

 

 

But what if radiation is fungible?

The divergence between the regression diagnosed ‘sensitivity’ to changes in solar intensity and volcanic aerosols is both surprising and puzzling.   The divergence is reported (albeit to a smaller extent) in the 1950 to 2012 regressions as well as the 1975 to 2012 regressions. In each case, the regression reports a best estimate response to solar cycle forcing which is more than twice as high as volcanic response on a watts/M^2 basis.   Lots of people expect the solar cycle to contribute to total forcing in a normal (fungible) way.  Figure 12 (from GISS) shows that for climate modeling, the folks at GISS think there is nothing special about solar cycle driven forcing.

For the diagnosed divergence between solar and volcanic sensitivities to be correct, there must be an additional mechanism by which the solar cycle substantially influences Earth’s temperatures, beyond the measured change in solar intensity.  I think convincing evidence of such a mechanism (changes in clouds from cosmic rays, for example) is lacking, although I am open to be shown otherwise.   But if we assume for a moment that there really is no significant difference in the response of Earth to solar and other radiative forcings, which seems to me plausible, then the above regression models ought to be modified by combining solar and volcanic into a single radiative forcing.

When the regressions are repeated for 1975 to 2012 with a single combined forcing (the sum of individual solar and volcanic) the minimum AIC score is for a quadratic secular function (rather than cubic when the two are independent variables), but the big change is that the regression diagnoses a longer lag for radiative forcing and a stronger response to radiative forcing (which is of course dominated by volcanic forcing).  Figure 13 shows the AIC scores and lag constants for polynomials of different orders when solar and volcanic forcings are combined.  The minimum AIC score with the combined forcings (quadratic secular function, 649.2) is slightly higher than the minimum for separate forcings (cubic secular function, 648.2), which lends some support to higher sensitivity for solar forcing.

Figure 14 shows the regression model and Hadley data, and Figure 15 shows the Hadley data adjusted for ENSO and combined solar and volcanic forcing.

 

 

The 1998 to 2012 slope is 0.0072 C/yr, while the 1979 to 1997 slope is 0.0165 C/yr; the recent trend is 44% of the earlier trend.

Why did F&R get different results?

The following appear to be the principle issues:

1.  Allowing volcanic and solar lags to vary independently of the others.

2.  Accepting physically implausible lags.

Treating solar and volcanic forcing as independent, combined with number 1 above seems to have some unexpected consequences.  Figure 16 shows the lagged and un-lagged volcanic forcing along with the un-lagged solar forcing.  The two major volcanic eruptions between1979 and 2012 (El Chichon and Pinatubo) happen to occur shortly after the peak of a solar cycle.

 

The solar and volcanic signals are partially ‘aliased’ by this coincidence (that is, both acting in the same direction at about the same time), while the decline in solar intensity following the solar cycle peak in ~2001 did not coincide with a volcano.  Since there was a considerable drop in rate of warming starting at about same time as the most recent solar peak, and since the regression can “convert” some of the cooling that was due to volcanoes into exaggerated solar cooling due to aliasing, the drop in the rate of warming after ~2000 can be ‘explained’ by the declining solar cycle after ~2001.  In other words, aliasing of solar and volcanic cooling in the early part of the 1975-2012 period, combined with free adjustment of ‘sensitivity’ to the two forcings independently, gives the regression the flexibility to increase sensitivity to the solar cycle by reducing the sensitivity to volcanoes, so that the best fit to an assumed linear secular trend corresponds to a larger solar coefficient.  Allowing the solar and volcanic forcing to act with different lags further increases the ability of the regression to increase solar influence and diminish volcanic influence.  All of which contributes to the F&R conclusion of “no change in underlying warming trend.”

Of course, the same aliasing applies to the regression for 1950 to 2012, but since there are more solar cycles and more volcanoes in the longer analysis, and those do not alias each other well, the regressions for the longer period reports a smaller difference in ‘sensitivity’ to solar and volcanic forcings.  For example, with an assumed linear secular trend (similar to F&R, but using one lag for both solar and volcanic forcings), the 1975 to 2012 regression coefficients are: volcanic = 0.1294, solar = 0.5445, while the best fit regression from 1950 (4th order polynomial secular function) the coefficients are: volcanic: 0.1346, solar = 0.3283.

It will be interesting to see how global average temperature evolves over the next  6-7 years as the current solar cycle passes its peak and declines to a minimum.  If F&R are correct about the large, lag-free influence of the solar cycle, this should be evident in the temperature record…. unless a major volcano happens to erupt in the next few years!

Conclusions

There is ample evidence that once natural variation from ENSO, solar cycles and volcanic eruptions is reasonably accounted for, the underlying ‘secular trend’ in global average temperatures remains positive.   But it is also clear that the best estimate of that secular trend shows a considerable decline in warming compared to the 1979 to 1997 period.  The cause for that decline in the rate of underlying warming is not known, but factors other than ENSO, volcanic eruptions, and the solar cycle are almost certainly responsible.

 

Postscript

In my last post I showed how the low pass filtered Nino 3.4 index correlates very strongly with the average temperature between 30N and 30S, and can account for ~75% of the total tropical temperature variation.   To check for the possible influence of ENSO (ENI) on temperatures outside the tropics, I first calculated the “non-tropic history” from the Hadley global history and Hadley tropical history (non-tropics = 2* global – tropics).  I then checked for correlation between the non-tropics history and the ENI (which is calculated from the low pass filtered Nino 3.4 index) at each monthly lag from 0 to 12 months.  Significant correlation is present starting at ~4 months lag through ~11 months lag, with the maximum correlation at 7 months lag from the ENI.  I then incorporated this 7-month lagged ENI as a separate variable in the regressions discussed above, and found very statistically significant correlation in all regressions.  The coefficient was remarkably consistent in all regressions, independent of the assumed secular trend function, at ~1/3 that of the global influence of ENI itself.  (The ENI coefficient itself was also remarkably consistent.)  The 7-month lagged ENI influence was significant at >99.9% confidence in all regressions.  The increased variable count was included in the calculation of AIC.

I used a simple dual-lag response model (two-box model) instead of a single lag response because land areas (and atmosphere) have much less heat capacity than ocean, and so react much more quickly to applied forcing than the ocean.  The reaction of land temperatures would be expected to be in the range of an order of magnitude faster based on relative (short term) heat capacities, which on a monthly basis makes the land response essentially immediate (lag less than a few months).   If land and ocean areas were thermally isolated, we would expect the very fast response of land to represent ~30% of the total, and the slower response of the ocean to represent ~70%.  That is, in proportion to the relative surface areas.  However, land and ocean are not thermally isolated, and the fast land response is reduced by the slower ocean response because heat exchanges between land and ocean pretty quickly.  Modeling this interaction would appear to be a non-trivial task, but I guessed that a simple approximation is to reduce the relative weightings of land and increase that of the ocean, and assumed 15% ‘immediate’ response and 85% lagged response.  The lag constants optimized in the regression applied to only 85% of the forcing; 15% of the forcing was considered essentially immediate.  The above appears to improve R^2 values a bit in nearly all regressions I tried, but does not impact the conclusions: the underlying secular trend appears lower from 1998 to 2012 than from 1979 to 1997.

 

Introduction

Foster & Rahmstorf (1) used a multiple regression model based on solar variation, volcanic aerosols, and ENSO to estimate how those factors have influenced surface temperature since 1979; the paper is basically a rehash, with some changes, of earlier published work by others (see for example http://www.agci.org/docs/lean.pdf and references). F&H adjusted measured changes in Earth’s surface temperature based on the results of their regression model, and claimed that the apparent slowdown in warming over the past 10+ years is entirely the result of natural variation, and that there has been absolutely no change in the underlying (secular) rate of warming since 1979. Oh yes, they also concluded that it is critical for people stop burning fossil fuels immediately…. though it is not immediately obvious how a multiple regression model on global temperatures leads to that conclusion.

Many people found the F&R paper to be technically weak, and its conclusions doubtful; my personal evaluation was that the paper was little better than a mindless curve-fit exercise. In spite of the coverage the paper got in some publications, I would normally prefer to ignore such things. But since the F&R paper seems to now have taken on the character of an urban legend, and is pointed to by warming-concerned folks whenever someone notes that warming has been much slower recently, I figured any reasoned critique of F&R is a useful endeavor.

F&R considered the influence of the solar cycle, a change of about 0.1% in solar intensity from peak to trough of the cycle, separately from the effects of stratospheric volcanic aerosols, even though both are expected to change the intensity of solar radiation reaching the Earth’s troposphere and surface. (Why should solar intensity change and volcanic aerosol forcing not be fungible?) Like some earlier publications, F&R also (strangely) limited their analysis to post 1979, even though data on volcanoes, solar cycles, and ENSO over longer periods is available. F&R concluded that variation in solar intensity has much greater influence on surface temperature, on a degree/watt/M^2 basis, than an equivalent change due to stratospheric volcanic aerosols, and further conclude the response of surface temperature to solar intensity variation is essentially instantaneous (no lag!), while stratospheric aerosols influence surface temperature only with considerable lag. Odd, very odd.

Here I offer what I believe is a more robust regression analysis of the same three variables (volcanic aerosols, ENSO, and solar cycle) on temperature evolution since 1950. I will show:

1) An improved index for accounting for ENSO.

2) The best regression fit is found when volcanic aerosols and solar intensity variation are lagged considerably due to thermal inertia of the system. The estimates for the influence of both (on a degrees/watt/M^2 basis) are very similar, not dramatically different.

3) After taking ENSO, volcanic aerosols, and solar cycles into account, the best estimate rate of warming from 1997 to 2012 is less than 1/3 the rate of warming from 1979 to 1996.

 

 

I. A Slightly Improved Method for Estimating ENSO Influence on Temperature Trends

The Nino 3.4 index is the monthly average temperature anomaly, in Celsius degrees, for the roughly rectangular area of the Pacific ocean bounded by 120 and 170 degrees west, 5 degrees north and 5 degrees south. This region represents only about 2.5% of the surface area of the Earth’s tropics (~30 north to ~30 south), yet is known to be strongly correlated with the ENSO and with changes in average temperature in the tropics. (For a more complete description see: http://www.ncdc.noaa.gov/teleconnections/enso/indicators/sst.php). Some months ago in a comment at The Blackboard, Carrick showed that Nino 3.4 shows little or no correlation, at any lag period, with temperatures outside of the tropics.  That is, ENSO strongly influences tropical temperatures but does not influence temperatures outside the tropics very much.  I concluded that if one is going to “account for” the influence of ENSO on global average temperatures using Nino 3.4, then it would be best to estimate the influence based on the variation in temperature anomaly for the tropics only. Eliminating uncorrelated temperature data from higher latitudes ought to improve signal to noise ratio, and yield a more accurate estimate of ENSO driven temperature changes.

The Nino 3.4 index, lagged two or three months, correlates reasonably well with temperature variation in the tropics, and can account for ~65% – 70% of the measured variation in average temperature. But can Nino 3.4 actually provide more information than gleaned from the 2 or 3 month lag correlation?

The answer seems to be that there is a bit more information available. If we consider ENSO to be a cyclical redistribution of heat that accumulates in the tropical Pacific, then it becomes clear the response to a change in ocean surface temperature in the Nino 3.4 region can’t be immediate, nor is the influence going to be accurately described by a specific lagged monthly Nino 3.4 value. During La Nina, stronger trade winds push warm surface water westward toward the Pacific warm pool, and that water is replaced with cooler water which upwells, mainly in the eastern Pacific. When the trade winds drop, an El Nino begins, with warm water flowing eastward from the Pacific warm pool, while the rate of upwelling in the eastern Pacific drops, which leads to warming in the eastern Pacific. The temperature response of the tropics ought to be something other than a simple lag of the Nino 3.4 index, since it takes time for heat to be distributed throughout the tropics.

So how can we use a direct measure of the tropical Pacific temperature anomaly (Nino 3.4) to better estimate the response of global average tropical temperature to ENSO? I reasoned as follows: The temperature rise in the tropics that is associated with an increasing Nino 3.4 temperature takes time to be distributed over all of the tropics, so any response should be gradual. As the tropical temperature rises, heat loss to space increases, and the warming influence for any single month should decay gradually to nothing. The influence of an instantaneous change (eg. a rise in the Nino 3.4 index from 0 C to 2C for only one month, followed by a flat Nino 3.4 index of 0 C for many months) ought to show an exponential-like decay from an initially strong influence.  There is not a single monthly Nino 3.4 influence at an optimal lag time, but rather a continuously evolving influence over some extended period. A strong El Nino or La Nina continues to have influence on tropical temperatures even after the Nino 3.4 index has returned to a neutral state.

I modeled the evolution of Nino 3.4 influence by iteratively calculating a new monthly index I call the “Effective Nino Index” (ENI):

ENI(n) = k * ENI(n-1) + (1-k) * Nino34d(n-1)

where ENI(n) is the Effective Nino 3.4 Index
n is the current month
(n-1) is the previous month
Nino34d(n-1) is the detrended Nino 3.4 index for the previous month
k is a constant between zero and one

ENI(n) is essentially a low pass filtered representation of all earlier Nino 3.4 values.   I tested several values of k to see what value generates an ENI which best correlates with temperature evolution in the tropics. Since ~1997, the temperature trend in the tropics has been relatively flat and not influenced by major volcanic eruptions, so I ran a regression of ENI against the detrended tropical temperature anomaly for 1997 to present (I used the Hadley Hadcrut4 tropics history, downloaded from Wood For Trees).  The best correlation between ENI and average tropical temperature is at k = 0.703. In other words, in any single month, the running history (2 and more months past, represented by ENI(n-1)) contributes 70.3% of the ENSO influence on average tropical temperature, and the previous month’s Nino 3.4 index contributes 29.7% of the influence on average tropical temperature. Figure 1 shows how the relative influence of any single month of Nino 3.4 declines over time, with zero months lag meaning the current month.  (Click on any image to view at the original resolution.)

A comparison of Nino 3.4 with ENI is shown in Figure 2. The lagging effect of the low-pass filter function is clear.  Please note that ENI is not a temperature index per se, but an index that represents the weighted contribution of all past Nino 3.4 temperatures, with the relative influence of earlier Nino 3.4 values falling rapidly in importance the further back in time you look.

Figure 3 shows the ENI and the detrended tropical temperature for 1997 to 2012 (Hadcrut4, downloaded from Wood for Trees) on the same graph, and Figure 4 shows the detrended tropical temperatures and ENI for 1950 to 2012.

There is very good correlation in the 1997 to 2013 period, where volcanic influences are minimal. You may note in figure 4 that periods of significant deviation between the detrended tropical temperature anomaly and the ENI correspond to the aftermath of major volcanic eruptions, which is consistent with significant aerosol cooling. The “adjusted” tropical temperature model based on the ENI regression against tropical temperatures is:

Tadj = Torg – (0.1959 +/- 0.016) * ENI

Where Torg is the original Hadley temperature anomaly for the tropics. +/-0.016 is the two sigma uncertainty for the model coefficient.

For the 1997 to 2012 period, the model’s F statistic was 594 (very highly significant), and the R^2 value was 0.756, meaning 75.6% of the total variance in tropical temperatures is predicted by the ENI value. It is important to note that ‘predicted’ is a suitable word, since the ENI influence is due to the combination of all earlier months’ Nino 3.4 values, not the present Nino 3.4 value.  Since the ENI is based on the detrended Nino 3.4 index, there is no net contribution to ENI from any general warming of the ocean surface over time.

Figure 5 shows the above ENI adjustment applied to all the Hadcrut4 tropical temperature data since 1950. As we might expect, the influence of volcanic aerosols from Pinatubo shows up much more clearly than in the unadjusted temperature data.

I will use the ENI in the combined regression analysis that includes volcanic and solar effects.

 

II. About Those Natural Forcings

NASA GISS provides data for their estimate of aerosol influences from 1850 to present (http://data.giss.nasa.gov/modelforce/strataer/). The data is in the form of Aerosol Optical Depth (AOD at 550 nm wavelength), which is converted into a net forcing value (watts/M^2) by multiplying the AOD by a constant of 23 (NASA’s value). The GISS volcanic aerosol forcing since 1950 is shown in Figure 6.

Direct measurements of solar intensity over the solar cycle are only available since 1979 (via satellites), but the correlation between sunspot number (SSN) and measured changes in solar intensity is good, so it is possible to estimate the historical variation in solar intensity based on SSN records. To estimate solar intensity variation, I used the following empirical equation, which comes from regressing measured solar intensity with sunspot number (data from a spreadsheet by Leif Svalgaard):

Solar intensity = 1365.45 + (0.006872 * SSN)  watts/M^2

Where solar intensity is measured above the Earth’s atmosphere, and SSN is the monthly sunspot number. The R^2 for this regression was 0.984.  The variation of solar intensity about an average value is then:

Variation = 0.006872 * (SSN – AvgSSN) watts/M^2

Where AvgSSN is the average number of sunspots over the period being studied (in this case from 1950 to 2012).

If we assume Earth’s albedo is 30%, and average over the entire surface (a factor of 4 compared to the cross-sectional area Earth presents to the Sun), the variation in solar energy reaching the Earth (including the troposphere) is:

Variation = (0.7/4) * 0.006872 * (SSN – AvgSSN) = 0.001203 * (SSN – AvgSSN) watts/M^2

Since the solar cycle is ~11 years long, we expect solar forcing to generate a temperature response with peaks separated by ~11 years. Figure 7 shows the calculated solar forcing since 1950.

 

 

III. Regression Model

The regression model has three independent variables: the ENI, with nominal units of temperature (as described above), lagged volcanic forcing, and lagged solar cycle forcing (both with nominal units of watts/M^2). We do not expect an instantaneous temperature response to volcanic and solar forcing, since the thermal mass of the Earth’s atmosphere, land surface and ocean surface are expected to slow the response… that is, to introduce lag between the applied forcing and the response.

A very accurate estimate of the global temperature response to solar and volcanic forcing history would require an accurate model of ocean heat uptake at different latitudes over time, as well as an accurate model of heat transport between high and low latitudes, between land and ocean, and between Earth and space. Since this type of model arguably doesn’t exist, I am forced to use a much simpler lag-type model. The lag model is based on a single constant value with a repetitive monthly calculation that approximates a low pass filter function:

EF(n) = EF(n-1) * (1 – K) + F(n) * K

where:
EF(n) is the effective forcing for month n (solar or volcanic)
F(n) is the actual forcing for month n (solar or volcanic)
K is a decay constant

When K =1, the effective forcing is identical to the actual current forcing. Smaller values of K introduce increasing lag in the response. This type of function is essentially equivalent to the expected response of a “slab” type ocean, or to Lucia’s ‘Lumpy’ model response.  Please note that the lag applies to both solar and volcanic aerosol forcings, since these are both radiative forcings.

Since I did not a priori know the best value of K, I tried different values of K and found the value which gave the best fit regression (that is, the highest R^2 value) for the three variables against the detrended monthly Hadley temperature series from 1950 to 2012. The best fit for K was 0.031. Figure 8 shows the “step response” of the lag function with K = 0.031, with F(n) starting at zero and then stepping to constant value of 1 at month 1.

Detrending of the temperature series was used prior to regression because the underlying long-term secular trend, whether due to GHG forcing alone or in combination with other long term influence(s), can’t be accurately modeled by the three variables in the regression, since these three variables are all expected to have relatively short term influence. Using the original temperature data (not detrended) distorts the regression fit by essentially forcing the regression to explain all the temperature change, including any slow secular trend, using the three short-influence variables, and so yields very poor (even physically nonsensical) results.

Figure 9 shows the original and lagged volcanic aerosol forcing, and Figure 10 shows the original and lagged solar forcing.

The best fit regression (with K = 0.031) yields the following constants:

ENI:         0.1099 +/-0.0118 (+/- 2-sigma uncertainty)
Volcanic: 0.2545 +/- 0.0277
Solar:       0.233 +/- 0.231

R^2 for the regression was 0.445 (44.5% of the variance was accounted for by the model).

The much greater uncertainty in the solar influence is due to the solar forcing being quite small compared to the other two. Still, it is encouraging that the regression shows the best estimates for response to both radiative forcing variables are very similar… just as one might expect, since radiation is fungible.

Figure 11 shows the temperature influence of the three variables and their combined influence based on the regression constants for each.

Figure 12 shows an overlay of the detrended Hadley temperature series and the sum of the three adjustments (both offset to average zero, which makes visual comparison easier), and Figure 13 shows the adjusted and unadjusted Hadley global temperature series.

I have added the slope lines for the adjusted series from 1979 to 1996 (inclusive) and from 1997 to 2012 (inclusive). The slope since 1997 is less than 1/6 that from 1979 to 1996.

 

IV. Comments, Conclusions, Caveats, and Uncertainties

Warming has not stopped, but it has slowed considerably. This analysis can’t prove the cause for that change in rate of warming, but any suggestion that solar cycles, volcanic aerosols, and ENSO are completely responsible for the recent slower warming rate is not supported by the data. Some may suggest long term cyclical variation in the secular warming rate has caused the recent slow-down, but this analysis can’t support or refute that suggestion.

It is encouraging that the influence of the ENI on global temperatures (as calculated by the by the global regression analysis) is just slightly more than half the influence found for the tropics alone (30S to 30N): 0.1099+/- 0.0118 global versus 0.1959+/-0.016 tropics. Since Carrick showed almost no correlation of ENSO with temperatures outside the tropics, and since 30S to 30N represents exactly half the Earth’s surface, we could reasonably expect the regression constant for the entire globe to be about half as large as for the tropics… and it is indeed very close to half (and within the calculated uncertainty limits).

The analysis indicates that global temperatures were significantly depressed between ~1964 and ~1999 compared to what they would have been in the absence of major volcanoes.

Here are a few caveats and uncertainties. First, the analysis is only as good as the data that when into it. Historical volcanic forcing from GISS is at best an estimate for all eruptions before Pinatubo; if the GISS volcanic forcing is wrong, then this could distort the regression results. The same is true for all other data, including the Hadley temperature series and the sunspot number model used to calculate solar forcing. While sunspot number is an excellent proxy for solar intensity over the last 3 solar cycles, that does not guarantee sunspot number has always been an equally excellent proxy for solar intensity.

Second, the single constant low-pass filter function used to calculate lagged solar and volcanic forcings is a fairly crude representation of reality. While the true lag function is almost certainly similar in shape, it will not be identical, and this too could distort the regression analysis to some extent. The reality is that there are a multitude of lag constants associated with heat transfer to/from different locations, especially different depths of the ocean.

Third, it is tempting to infer very low climate sensitivity from the regression constants for volcanic aerosols and solar cycle forcing (these constants have units of degrees/watt/M^2, and the values correspond to a climate sensitivity of a little less than 1C per doubling of CO2). This temptation should be resisted, because the model does not consider the influence of (slower) heat transfer between the surface and deeper ocean. In other words, the calculated impact of solar and volcanic forcings would be larger (implying somewhat higher climate sensitivity) if a better model of heat uptake/release to/from the ocean were used.

 

Request for only constructive comments:  Skydragon slayers and rabid catastrophic warmers should not feel their comments are required or requested.

 

 

(1) Grant Foster and Stefan Rahmstorf 2011 Environ. Res. Lett. 6 044022

An important issue for the future is the potential for significant sea level rise due global warming. Sea level rise consists of two parts: a steric contribution (volume increase due to an increase in average ocean temperature, which reduces average seawater density), and a mass contribution (volume increase due mainly to melting of land supported ice… glaciers, ice caps, and ice sheets). It has been projected by some (Vermeer & Rahmstorf, and others) that the mass contribution will accelerate rapidly in the next few decades as average surface temperature increases; these projections suggest a rapidly accelerating sea level rise, starting very soon. While the IPCC AR4 projections of sea level rise through 2100 are modest, some more recent projections suggest increases of well over 1 meter by 2100. My guest posts at the Blackboard in July 2011 addressed this issue, and the results of my model suggest future sea level increases in reasonable agreement with the AR4 projections, for a wide range of assumed rates for surface temperature increase.

The steric contribution is clearly defined if warming of the ocean is known. The ARGO system of floats measures ocean temperatures between the surface and 2,000 meters depth, with the highest quality data between the surface and 700 meters. ARGO became fully operational in ~2003. But some earlier ocean heat data exists, with the best data set (Levitus et al) dating from 1955. The mass contribution to sea level rise since 1955 is less certain, due in part to uncertainty in the available tide gauge data. However, accurate satellite measurements of total sea level rise are available from 1993 to present (~18 years).

The graph below shows the satellite measurements of sea level, with seasonal variation removed. Seasonal variation in sea level is caused by seasonal accumulation of precipitation on land (ice, water) as well as seasonal variation in total ocean heat content (seasonal steric contribution).

Note that there has been an apparent reduction in rate of sea level rise starting ~2003-2004. What caused this reduction? It is well known that ocean surface temperatures have not risen as much over the last decade as in the 2 decades before, and this has reduced the rate of heat accumulation in the oceans. So a reasonable expectation is lower steric contribution. The regularly updated ocean heat data from Levitus et al (http://www.nodc.noaa.gov/OC5/3M_HEAT_CONTENT/) in the graph below, shows that the 0-700 meter steric contribution fell to near zero starting in 2003.

If you subtract the 0-700 meter steric contribution from the total sea level change, the net is an estimate of the mass contribution alone, that is, the contribution mostly from the melting of land supported ice. The graph below shows the satellite data with the steric contribution removed.

Note that the steric data is only available as yearly averages, while the satellite data points represent a few weeks each, so subtracting one form the other I had to use annual average trends for the steric contribution.

There are a couple of interesting things in this graph. First, it is clear that once the change in steric contribution is accounted for, the trend in the remaining data (that is, mostly ice melt contribution) has remained relatively constant for the last 18 years, at about 2.3 mm per year. There is no indication of an acceleration (nor deceleration) in melt contribution over that 18 years. Second, the influence of the ENSO becomes more obvious once the 0-700 steric contribution is removed: there are clear positive and negative deviations from the overall (linear) trend corresponding to the 1998 and 2010 el Ninos and the 2008 la Nina. In fact, it looks like much of the overall remaining variation correlates quite well with an ENSO index like Nino 3.4. This suggests that either a) short term variation in average temperature is causing small but rapid changes in snow/ice melt contribution, b) there is short term ocean heat content change which is not captured well by the Levitus data, or c) a combination of these effects.

The graph below shows the 0-700 meter steric adjusted satellite sea level compared to the Nino 3.4 index; the heavy blue line is an 11 point centered moving average filter. The Nino 3.4 index was multiplied by a factor of 2.5 and a linear slope was added to match that of the satellite data. Most (but not all) variation from a linear trend in the satellite sea level data is matched by the Nino 3.4 index. There is essentially no time offset between the Nino 3.4 variation and the seal level variation, unlike global average surface temperature, which tends to lag 3-6 months behind changes in the Nino 3.4 index. The lack of lag suggests that short term changes in ocean heat content is the most likely explanation for the correlation, rather than rapid changes in melting of ice/snow.

Conclusions, Observations, Cautions
The satellite sea level data shows no evidence of acceleration in the rate of ice melt over the past 18 years, and the observed reduction in the rate of sea level rise since ~2003 is consistent with a much reduced rate of ocean heat accumulation. The relatively short period of the satellite record means that one should be cautious about drawing too many conclusions from the data; the measured trend of the last 18 years could be influenced significantly by unaccounted factors. The good news is that continued satellite measurements of sea level and ARGO measurements of ocean heat content will allow more confident estimates of any change in the long term trend over the next decade or two. I personally expect the sea level trend to not accelerate nearly as much as some have suggested, and I expect that projections of extreme rise in sea level by 2100 will be largely refuted within 15- 20 years.

Any accumulation of heat below 700 meters is not included in the Levitus et al data, though it is reasonable to believe that at least some deeper accumulation is taking place (and there have been some estimates which range up to ~0.35 watt/M^2). Therefore, the contribution due to mass increase is almost certainly somewhat lower than 2.3 mm/year. 0.35 watt/M^2 of “deep” accumulation would be contributing up to ~ 0.4 mm per year to total expansion, so assuming 0.35 watt/M^2 deep heat accumulation, the net mass contribution to sea level rise is likely to be somewhere near ~1.9 mm per year. This is consistent with Cazenave et al (2008), who concluded (based on GRACE satellite measurement of ocean mass) that the mass increase in the ocean between 2003 and 2008 was 1.9+/- 0.1 mm/year.

The satellite sea level data is adjusted to compensate for estimated growth in ocean basin volume (about +0.3 mm per year is added to the measured level) due to slow glacial rebound in regions where the heavy ice sheets of the last ice age depressed the local surface by hundreds of meters. But the satellite data is not adjusted to account for potential accumulation of water in man-made reservoirs, which reduces the measured rate of rise, nor for pumping of groundwater (‘ancient water mining’), which increases the measured rate of rise. Recent estimates of these contributions are a significant fraction of the measured change in sea level. The graph below (from one of my earlier posts) shows my rather crude graphical presentation of two published estimates of these factors (Wada et al and Chao et al).

The estimated rate of reservoir accumulation was more rapid between ~1955 and ~1985, yielding a net year on year removal of water from the oceans, but from 1985 to present, the estimated rate of groundwater pumping has been greater than reservoir accumulation, yielding a net year-on-year addition of water to the oceans. The cumulative net contribution year-on-year since 1993 is about +7 mm, or approximately 0.39 mm per year. In other words, during the satellite period, the true overall sea level change due to warming could be significantly lower than the measured trend; ~2.71 mm/year instead of ~3.1 mm/year. (For reference: the average rate of rise since 1910, based on tide gauges, is about 1.9 – 2.0 mm per year.)

Combining the potential effects of deep ocean heat accumulation, reservoir accumulation, and ground water pumping, the true average melt contribution during the satellite period may be as low as: 2.3 – 0.4 – 0.4 = 1.5 mm per year, while the true average steric contribution for that period may be as much as 1.2 mm per year.

References:

Global ocean heat content 1955–2008 in light of recently revealed
instrumentation problems; Levitus et al, Geophysical Research Letters (2009)

Global sea level linked to global temperature; Martin Vermeer and Stefan Rahmstorf, PNAS (2009)

Sea level budget over 2003–2008: A reevaluation from GRACE space gravimetry, satellite altimetry and Argo; Cazenave et al, Global and Planetary Change (2008)

Global depletion of groundwater resources; Wada et al, Geophysical Research Letters (2010)

Impact of Artificial Reservoir Water Impoundment on Global Sea Level; Chao, et al., Science (2008)