Problem 1 of Climate Science

Problem 1. If temperature is adequately represented by a deterministic trend due to increasing GHGs, why be concerned with the presence of a unit root?

Rather than bloviate over the implications of a unit root (integrative behavior) in the global temperature series, a more productive approach is to formulate an hypothesis, and test it.

A deterministic model of global temperature (y) and anthropogenic forcing (g) with random errors e is:

An autoregressive model of changes in temperature Δyt uses a difference equation with a deterministic trend and the previous value of y or yt-1:


Written this way, the presence of the unit root in an AR1 series y is equivalent to the coefficient c equaling zero (see

I suspect the controversy can be reduced to two simple hypotheses:

H0: The size of the coefficient b is not significantly different from zero.
Ha: The size of the coefficient b is significantly different from zero.

The size of the coefficient should be indicative of the contribution of the deterministic trend (in this case anthropogenic warming) to the global temperature.

We transform the global temperature by differencing (an autoregressive or AR coordinate system), and then fit a model just as we would with any model.

In the deterministic coordinate system, b is highly significant with a strong contribution from AGW. For the AGW forcing I use the sum of the anthropogenic forcings in the RadF.txt file W-M_GHGs, O3, StratH2O, LandUse, and AIE.


Call: lm(formula = y ~ g)
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.34054 0.01521 -22.39 <2e-16 ***
g 0.31573 0.01802 17.52 <2e-16 ***

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.1251 on 121 degrees of freedom
Multiple R-squared: 0.7172, Adjusted R-squared: 0.7149
F-statistic: 306.9 on 1 and 121 DF, p-value: < 2.2e-16

The result is very different in the AR coordinate system. The coefficient of y is not significantly greater than zero (at 95%) and neither is b.


Call: lm(formula = d ~ y + g + 0)
Estimate Std. Error t value Pr(>|t|)
y -0.06261 0.03234 -1.936 0.0552 .
g 0.01439 0.01088 1.322 0.1887

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.101 on 121 degrees of freedom
Multiple R-squared: 0.0389, Adjusted R-squared: 0.02302
F-statistic: 2.449 on 2 and 121 DF, p-value: 0.09066

Perhaps the main contribution of AGW is since 1960, so we restrict the data to this period and examine the effect. The deterministic trend in AGW is greater, but still not significant.


Call: lm(formula = d ~ y + g + 0)
Estimate Std. Error t value Pr(>|t|)
y -0.24378 0.10652 -2.289 0.0273 *
g 0.03050 0.01512 2.017 0.0503 .

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.1149 on 41 degrees of freedom
Multiple R-squared: 0.1284, Adjusted R-squared: 0.08591
F-statistic: 3.021 on 2 and 41 DF, p-value: 0.05974

But what happens when we use another data set. Below is the result using GISS. The coefficients are significant but the effect is still small.


> Prob1(GISS,GHG)
Call: lm(formula = d ~ y + g + 0)
Estimate Std. Error t value Pr(>|t|)
y -0.27142 0.06334 -4.285 3.69e-05 ***
g 0.06403 0.01895 3.379 0.00098 ***

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.1405 on 121 degrees of freedom
Multiple R-squared: 0.1375, Adjusted R-squared: 0.1232
F-statistic: 9.645 on 2 and 121 DF, p-value: 0.0001298

So why be concerned with the presence of a unit root? It has been argued that while the presence of a unit indicates that using OLS regression is wrong, this does not contradict AGW because the effect of greenhouse gas forcings can still be incorporated as deterministic trends.

I am not 100% sure of this, as the differencing removes most of the deterministic trend that could be potentially explained by g.

If the above is true, there is a problem. When the analysis respects the unit root on real data, the deterministic trend due to increasing GHGs is so small that the null hypothesis is not rejected, i.e. the large contribution of anthropogenic global warming suggested by a simple OLS regression model is a spurious result.

Here is my code. Orient is a functions that matches two time series to the same start and end date.

Prob1<-function(y,g) {

Australian Temperature Adjustments

If you have seen the articles on the NZ temperature adjustments and Nordic temperature adjustments you might be interested in the Australian data.

The issue with NZ and Nordic data that the raw temperature data for weather stations do not show the temperature increases indicated by the IPCC, raising the question of how the data have been adjusted.

As Prof. Karlen states in the ClimateGate email #1221683947, temperature at many stations has not exceeded early 20th century temperatures:

.. data sets show an increase after the 1970s to the same level as in the late 1930s or lower. None demonstrates the distinct increase IPCC indicates.

Here is the plot of means of Australian raw data for 103 temperature stations, based on the file downloaded from the Australian BoM web site and collated by Steve McIntyre.


Continue reading Australian Temperature Adjustments

Is OHC Accelerating?

Code and figures to quantify the answer to the question “Is ocean heat content is accelerating?” are below. The idea is that ‘acceleration’ is synonymous with the significance of a quadratic term in a regression:

1. Annual OHC data from NODC.

2. Fit a regression model (M1) incorporating linear and periodic terms of period 60 years (to account for Pacific Decadal Oscillation):

M1 = lm(OHC~x+sin(f)+cos(f))

3. Fit another regression model with the addition of a quadratic term,

M2 = lm(OHC~x+sin(f)+cos(f)+I(x^2))

4. Compare the reduction in the regression sum of squares due to the incorporation of the quadratic term, taking into account the loss of degrees of freedom due to autocorrelation (see for tests of nested models)

The result below shows M1 as a solid line and M2 as a dashed line. The p value for the F test is a marginally significant 0.052 (not significant at the 95% CL) for an improvement in the model due to addition of a quadratic term.


Continue reading Is OHC Accelerating?

Comment on McLean et al Submitted

Here is the abstract for our comment submitted to Geophysical Research Letters today. Bob Tisdale is acknowledged as the source of the idea in the first paragraph. Lets see how it goes. If you would like a copy, contact me via the form above.

Update: Now available from arXiv

Comment on “Influence of the Southern Oscillation on tropospheric temperature” by J. D. McLean, C. R. de Freitas, and R. M. Carter

David R.B. Stockwell and Anthony Cox


We demonstrate an alternative correlation between the El Nino Southern Oscillation (ENSO) and global temperature variation to that shown by McLean et al. [2009]. We show 52% of the variation in RATPAC-A tropospheric temperature (and 59% of HadCRUT3) is explained by a novel cumulative Southern Oscillation Index (cSOI) term in a simple linear regression model and 65% of RATPAC-A variation (67% of HadCRUT3) when volcanic and solar effect terms are included. We review evidence from physical and statistical research in support of the hypothesis that accumulation of the effects of ENSO can produce natural multi-decadal warming trends. Although it is not possible to reliably determine the relative contribution of anthropogenic forcing and SOI accumulation from multiple regression models due to collinearity, these results suggest a residual accumulation of around 5 ± 1% and up to 9 ± 2% of ENSO-events has contributed to the global temperature trend.

Continue reading Comment on McLean et al Submitted

Weekly Roundup

Last week I prepared a comment defending McLean’s SOI paper. I shall send it in shortly. Basically, I extend their analysis a little and show that the majority of variation in a linear regression model predicting global temperature (not differences) can be accounted for using SOI-related terms.

The sorry-state of Dr. Hathaway’s prediction record is in the news.

Continue reading Weekly Roundup

Influence of the Southern Oscillation on tropospheric temperature

A potential AGW buster, attributing decadal temperature variation largely to internal oceanic effects, ENSO and over the longer term the 1976 Great Pacific Climate Shift, as we did here, is a new paper by Australian John McLean, with New Zealander Chris de Freitas, and Australian ex-pat Kiwi Bob Carter.

That mean global tropospheric temperature has for the last 50 years fallen and risen in close accord with the SOI of 5–7 months earlier shows the potential of natural forcing mechanisms to account for most of the temperature variation.

While the bottom line of this paper is that the change in SOI accounts for 72% of the variance in global temperature for the 29-year-long MSU record and 68% of the variance in global temperature for the longer 50-year RATPAC record, I think the claim of a longer term temperature effect could have been better supported. They stated:

Lean and Rind [2008] stated that anthropogenic warming is more pronounced between 45°S and 50°N and that no natural process can account for the overall warming trend in global surface temperature. We have shown here that ENSO and the 1976 Great Pacific Climate Shift can account for a large part of the overall warming and the temperature variation in tropical regions.


However, the assertion comes down to Figure 4 where they identify that the mean of the SOI (and temperature) seems to change at 1976. This model is not identified rigorously with any analysis, but is stated as an observation in the text.

Continue reading Influence of the Southern Oscillation on tropospheric temperature

A semi-empirical approach to sea level rise

Published in Science, this Rahmstorf 2007 article provides a high-end estimate of sea level rise of over a meter by the end of the century (rate of 10mm/yr). Linear extrapolation puts the rate of increase at only 1.4mm and 1.7mm per year depending on start date (1860 or 1950).

The paper was followed by two critical comments, both bashing the statistics, and these are attached to the link above. Rahmstorf replied to those comments. The issues raised are familiar to readers of this, CA, Lucia, and other statistical blogs: significance, autocorrelation, etc. and worth a read.

Worthwhile as the comments are, they do not look into the problem of the end-treatment used by Rahmstorf, and I look at that here.

All of the papers projecting these high end rates, and they all depend on the assumption of recent ‘acceleration’ in sea levels. That is, seem to depend on the rate of increase getting faster and faster.

Rahmstorf 2007 paper uses the smoothing method most recently savaged at CA here, where it was shown despite all the high-falutin’ language to be equivalent to a simple triangular filter of length 2M, padded with M points of slope equal to the last M points. My main concern is that at this crucial end-section, the data has been duplicated by the padding, effectively increasing the number of data points of very high slope.

The figure below shows a replication of the Rahmstorf smoothing with and without padding (moved down for clarity) (code below). Two sea level data sets are shown, one by Church “A 20th century acceleration in global sea level rise” (used in Rahmstorf, data available from CSIRO here) another by Jevrejeva “Recent global sea level acceleration started over 200 years ago?” (data here)

It should be noted this data ends in 2001-2, a truncation bound to maximize recent temperature increases.


Continue reading A semi-empirical approach to sea level rise

Preprint on climatic regime shifts

Download: Structural break models of climatic regime-shifts: claims and forecasts

Anthony asked if it would be difficult to statistically justify the breaks in temperature between 1976 and 1979 proposed by Quirk (2009) for Australian temperature. He has an interest in oceanographic regime-shifts and climate change. Sure, I said, a simple Chow test.

We ended up rebutting the Easterling & Wehner (2009) claim that describing temperatures since 1998 as declining is ‘cherry picking’, finding a major regime shift occurred in 1997, statistically justifying the use of 1997 as a starting point for temperature trends.

A regime-shift based temperature forecast follows logically from identification of significant breaks. Our paper, “Structural break models of climatic regime-shifts: claims and forecasts“, has been submitted to the International Journal of Forecasting, and is downloadable from arXiv.


Continue reading Preprint on climatic regime shifts

Recent Climate Observations: Disagreement With Projections

Appearing in Energy and Environment (ee-20-4_7-stockwell2) is a note by myself on a paper by IPCC lead authors Rahmstorf, S., Cazenave A., Church J.A., Hansen J.E., Keeling R.F., Parker D.E., and R.C.J. Somerville, Recent climate observations compared to projections published in Science in 2007.

As shown by 102 citations in Google Scholar already, Rahmstorf et al 2007 has been one of the main references for alarmist calls to action because the “climate system is responding more quickly than the climate models indicate”. Taking the first one off Google:

The strong trends in climate change already evident, the likelihood of further changes occurring, and the increasing scale of potential climate impacts give urgency to addressing agricultural adaptation more coherently. There are …

Adapting agriculture to climate change –, SM Howden, JF Soussana, FN Tubiello, N Chhetri, M … Proceedings of the National Academy of Sciences, 2007 – National Acad Sciences.

Respected on-line authors like Peter Gallagher, Mark Lawson and Lucia were concerned with the paper. Lucia attacked the ‘slide and eyeball’ approach. I engaged with Rahmstorf at RealClimate and wrote a number of articles on the uncertainty, until he told me in effect to ‘sod off and publish’. But rather than try to diagnose a sloppy methodology and be ignored, time and evidence has done the job instead. Here is my abstract.

Abstract: The non-linear trend in Rahmstorf et al. [2007] is updated with recent global temperature data. The evidence does not support the basis for their claim that the sensitivity of the climate system has been underestimated.

Its gratifying to read that the authors of the Copenhagen Synthesis Report do not seem to agree with Rahmstorf et al 2007 either, in reference to analysis in a figure that ostensibly used the same method as Rahmstorf et al 2007.

Figure 3 … shows the long-term trend of increasing temperature is clear and the trajectory of atmospheric temperature at the Earth’s surface is proceeding within the range of IPCC projections.


Continue reading Recent Climate Observations: Disagreement With Projections

Proof of AGW

Given the way the components of the surface temperature record extracted from the SSA (singular spectrum analysis) line up with various potential causes of climate change in the previous post here, the temptation is to latch onto series 2, and say, aha, there is the forcing due to increase in CO2. It’s the right shape, exponential. Its the right size, about 0.6C.

But looking into fractal data is like seeing pictures in clouds. Be suspicious of magic methods that pull explanations out of the air. Below I have plotted SSA decompositions of the the monthly global temperature anomaly from the HadCRU dataset from 1976 to the present, the period of most recent rise, and attributed largely to GHGs. Kind of zooming in.


Continue reading Proof of AGW

R code for SSA example

The code for plotting the non-linear temperature trend, using SSA (singular spectrum analysis) in the figure below is here – ssa-demo. I have made it as turnkey as I can. The steps are:

1. Get and Install package ssa ( I had to hand-compile and move the C shared library around so it would find it, not sure why.

2. Run the script below with source(“filename”). Uncomment line indicated after the first time to speed it up. You should get the following replication of the Rahmstorf figure with 11 and 14 embedding period.


Continue reading R code for SSA example

Plimer Review: Backpackers Guide to Global Warming

My signed copy of Heaven+Earth: Global Warming, the missing science by Australia’s most eminent geologist arrived last week. Peter Gallagher has just reviewed it here, and I agree with most of his impressions.

Apart from anything else it seems like a useful compendium of “History, Sun, Earth, Water, Air” facts and references, with over 500 pages and 2311 references. My hardcover version is laid out in a small book format with over-narrow margins, making it look a little like a “Backpackers Guide”. The language is quite informal too, so the impression fits.

Continue reading Plimer Review: Backpackers Guide to Global Warming

40 Years of Some BoM Australian Rural Temperature Data

The Australian Bureau of Meteorology (BOM) has acquired daily weather data from many sites for many years and compiled it into a base that compares well with that from other countries.

The data in various stages of treatment is available from Dr Stockwell through this site. It includes a small amount of infilling of missing data, usually by inserting the value(s) of an adjacent day. The infilling is not considered to alter the conclusions, but it is a mathematical convenience. It can be said of these 17 rural sites that –
•    the most northerly half averaged similar slopes to the southerly half
•    airport locations were similar to non-airport locations
•    there is insignificant correlation with nearby town populations
•    there does not seem to be a UHI effect
•    in some places, instrumental problems might be confused with climate responses
•    the slope of inland sites was greater by far than the slope of coastal sites.

The two purposes of this note are to solicit suggestions on why inland sites (which equate to higher elevations above sea level) mostly have higher slopes than coastal sites; and to make known the availability of worked data as outlined. Remember that the problem needing explanation is not simply moderation of temperatures by the sea. That might reduce data scatter, but it would not easily reduce temperature increase as happens inland.

Continue reading 40 Years of Some BoM Australian Rural Temperature Data

The value of tau

Admin: Posted up for Steve, with an initial response by Miklos. The slides Steve referred to are here. My bad for not telling Miklos that.

Link to TF&K08

Miskolczi theory proposes a tau (Ta if you will) significantly different from that found by at least a dozen other studies published in the peer-reviewed literature over more than a decade, as well as a number of other new relations A_A = E_D, f = 2/3 etc., etc.
Continue reading The value of tau

GBR recovery

A feel-good story of nature’s resiliency, “Doom and Boom on a Resilient Reef: Climate Change, Algal Overgrowth and Coral Recovery” has been making press with headlines focusing on the state of mind of the authors:

Marine scientists say they are astonished at the spectacular recovery of certain coral reefs in Australia’s Great Barrier Reef Marine Park from a devastating coral bleaching event in 2006.

Continue reading GBR recovery

Niche Modeling. Chapter Summary

Here is a summary of the chapters in my upcoming book Niche Modeling to be published by CRC Press. Many of the topics have been introduced as posts on the blog. My deepest thanks to everyone who has commented and so helped in the refinement of ideas, and particularly in providing motivation and focus.

Writing a book is a huge task, much of it a slog, and its not over yet. But I hope to get it to the publishers so it will be available at the end of this year. Here is the dustjacket blurb:

Through theory, applications, and examples of inferences, this book shows how to conduct and evaluate ecological niche modeling (ENM) projects in any area of application. It features a series of theoretical and practical exercises in developing and evaluating ecological niche models using a range of software supplied on an accompanying CD. These cover geographic information systems, multivariate modeling, artificial intelligence methods, data handling, and information infrastructure. The author then features applications of predictive modeling methods with reference to valid inference from assumptions. This is a seminal reference for ecologists as well as a superb hands-on text for students.

Part 1: Informatics

Functions: This chapter summarizes major types, operations and relationships encountered in the book and in niche modeling. This and the following two chapters could be treated as a tutorial in the R. For example, the main functions for representing the inverted ‘U’ shape characteristic of a niche — step, Gaussian, quadratic and ramp functions – are illustrated in both graphical from and R code. The chapeter concludes with the ACF and lag plots, in one or two dimensions.

Data: This chapter demonstrates how to manage simple biodiversity databases using R. By using data frames as tables,
it is possible to replicate the basic spreadsheet and relational database operations with R’s powerful indexing functions.
While a database is necessary for large-scale data management, R can eliminate conversion problems as data is moved between systems.

R and image processing operations can perform many of the
elementary spatial operations necessary for niche modeling.
While these do not replace a GIS, it demonstrates that generalization of arithmetic concepts to images can be implemented simple spatial operations efficiently.

Part 2: Modeling

Theory: Set theory helps to identify the basic assumptions
underlying niche modeling, and the relationships and constraints between these
assumptions. The chapter shows the standard definition of the niche as
environmental envelopes is equivalent to a box topology. It is proven that when
extended to infinite dimensions of environmental variables this definition
loses the property of continuity between environmental and geographic spaces.
Using the product topology for niches would retain this property.

Continue reading Niche Modeling. Chapter Summary