{"id":22623,"date":"2013-06-12T20:10:51","date_gmt":"2013-06-13T02:10:51","guid":{"rendered":"http:\/\/rankexploits.com\/musings\/?p=22623"},"modified":"2013-06-12T20:46:56","modified_gmt":"2013-06-13T02:46:56","slug":"estimating-the-underlying-trend-in-recent-warming","status":"publish","type":"post","link":"https:\/\/rankexploits.com\/musings\/2013\/estimating-the-underlying-trend-in-recent-warming\/","title":{"rendered":"Estimating the Underlying Trend in Recent Warming"},"content":{"rendered":"

Introduction <\/strong><\/h2>\n

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.<\/p>\n

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.<\/p>\n

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.<\/p>\n

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:<\/p>\n

1) An improved index for accounting for ENSO.<\/p>\n

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.<\/p>\n

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.<\/p>\n

 <\/p>\n

 <\/p>\n

I. A Slightly Improved Method for Estimating ENSO Influence on Temperature Trends<\/strong><\/h2>\n

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.\u00c2\u00a0 That is, ENSO strongly influences tropical temperatures but does not influence temperatures outside the tropics very much.\u00c2\u00a0 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.<\/p>\n

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?<\/p>\n

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.<\/p>\n

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.\u00c2\u00a0 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.<\/p>\n

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

ENI(n) = k * ENI(n-1) + (1-k) * Nino34d(n-1)<\/p>\n

where ENI(n) is the Effective Nino 3.4 Index
\nn is the current month
\n(n-1) is the previous month
\nNino34d(n-1) is the detrended Nino 3.4 index for the previous month
\nk is a constant between zero and one<\/p>\n

ENI(n) is essentially a low pass filtered representation of all earlier Nino 3.4 values.\u00c2\u00a0\u00c2\u00a0 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).\u00c2\u00a0 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.\u00c2\u00a0 (Click on any image to view at the original resolution.)<\/p>\n

\"Figure1\"<\/a><\/p>\n

A comparison of Nino 3.4 with ENI is shown in Figure 2. The lagging effect of the low-pass filter function is clear.\u00c2\u00a0 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.<\/p>\n

\"Figure2\"<\/a><\/p>\n

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.<\/p>\n

\"Figure3\"<\/a><\/p>\n

\"Figure4\"<\/a><\/p>\n

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:<\/p>\n

Tadj = Torg – (0.1959 +\/- 0.016) * ENI<\/p>\n

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

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.\u00c2\u00a0 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.<\/p>\n

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.<\/p>\n

\"Figure5\"<\/a><\/p>\n

I will use the ENI in the combined regression analysis that includes volcanic and solar effects.<\/p>\n

 <\/p>\n

II. About Those Natural Forcings<\/strong><\/h2>\n

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.<\/p>\n

\"Figure<\/a><\/p>\n

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):<\/p>\n

Solar intensity = 1365.45 + (0.006872 * SSN)\u00c2\u00a0 watts\/M^2<\/p>\n

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.\u00c2\u00a0 The variation of solar intensity about an average value is then:<\/p>\n

Variation = 0.006872 * (SSN – AvgSSN) watts\/M^2<\/p>\n

Where AvgSSN is the average number of sunspots over the period being studied (in this case from 1950 to 2012).<\/p>\n

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:<\/p>\n

Variation = (0.7\/4) * 0.006872 * (SSN – AvgSSN) = 0.001203 * (SSN – AvgSSN) watts\/M^2<\/p>\n

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.<\/p>\n

 <\/p>\n

\"Figure7\"<\/a><\/p>\n

 <\/p>\n

III. Regression Model<\/strong><\/h2>\n

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.<\/p>\n

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:<\/p>\n

EF(n) = EF(n-1) * (1 – K) + F(n) * K<\/p>\n

where:
\nEF(n) is the effective forcing for month n (solar or volcanic)
\nF(n) is the actual forcing for month n (solar or volcanic)
\nK is a decay constant<\/p>\n

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.\u00c2\u00a0 Please note that the lag applies to both solar and volcanic aerosol forcings, since these are both radiative forcings.<\/p>\n

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.<\/p>\n

\"Figure8\"<\/a><\/p>\n

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.<\/p>\n

Figure 9 shows the original and lagged volcanic aerosol forcing, and Figure 10 shows the original and lagged solar forcing.<\/p>\n

\"Figure9\"<\/a><\/p>\n

\"Figure10\"<\/a><\/p>\n

The best fit regression (with K = 0.031) yields the following constants:<\/p>\n

ENI: \u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0 0.1099 +\/-0.0118 (+\/- 2-sigma uncertainty)
\nVolcanic: 0.2545 +\/- 0.0277
\nSolar: \u00c2\u00a0 \u00c2\u00a0 \u00c2\u00a0 0.233 +\/- 0.231<\/p>\n

R^2 for the regression was 0.445 (44.5% of the variance was accounted for by the model).<\/p>\n

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.<\/p>\n

Figure 11 shows the temperature influence of the three variables and their combined influence based on the regression constants for each.<\/p>\n

\"Figure11\"<\/a><\/p>\n

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.<\/p>\n

\"figure<\/a><\/p>\n

\"Figure13\"<\/a><\/p>\n

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.<\/p>\n

 <\/p>\n

IV. Comments, Conclusions, Caveats, and Uncertainties<\/strong><\/h2>\n

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.<\/p>\n

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).<\/p>\n

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.<\/p>\n

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.<\/p>\n

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.<\/p>\n

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.<\/p>\n

 <\/p>\n

Request for only constructive comments:\u00c2\u00a0 Skydragon slayers and rabid catastrophic warmers should not feel their comments are required or requested.<\/p>\n

 <\/p>\n

 <\/p>\n

(1) Grant Foster and Stefan Rahmstorf 2011 Environ. Res. Lett. 6 044022<\/p>\n","protected":false},"excerpt":{"rendered":"

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 … Continue reading Estimating the Underlying Trend in Recent Warming<\/span> →<\/span><\/a><\/p>\n","protected":false},"author":13,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[15,5],"tags":[],"class_list":["post-22623","post","type-post","status-publish","format-standard","hentry","category-data-comparisons","category-global-climate-change"],"_links":{"self":[{"href":"https:\/\/rankexploits.com\/musings\/wp-json\/wp\/v2\/posts\/22623","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/rankexploits.com\/musings\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/rankexploits.com\/musings\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/rankexploits.com\/musings\/wp-json\/wp\/v2\/users\/13"}],"replies":[{"embeddable":true,"href":"https:\/\/rankexploits.com\/musings\/wp-json\/wp\/v2\/comments?post=22623"}],"version-history":[{"count":0,"href":"https:\/\/rankexploits.com\/musings\/wp-json\/wp\/v2\/posts\/22623\/revisions"}],"wp:attachment":[{"href":"https:\/\/rankexploits.com\/musings\/wp-json\/wp\/v2\/media?parent=22623"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/rankexploits.com\/musings\/wp-json\/wp\/v2\/categories?post=22623"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/rankexploits.com\/musings\/wp-json\/wp\/v2\/tags?post=22623"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}