Rahmstorf (2011): Robust or Just Busted (Part 3) New code for putting Rahmstorf to the test

August 19, 2012

This is part 3 of a multi-part series about “Testing the robustness of semi-empirical sea level projections,” Rahmstorf, et. al., Climate Dynamics, 2011. You can see an index of all parts here. I frequently refer to this paper as R2011.

I will refer to Stefan Rahmstorf’s ”Testing the robustness of semi-empirical sea level projections”  as R2011 [1].

Although I have written very little in this blog in recent weeks, I have been very active on the sea level rise data and code front. I have been putting together a method for consistent processing of temperature and sea level data according to the predominant Vermeer and Rahmstorf 2009 model (VR2009)[2].  This is important because Rahmstorf claims that his model is “robust,” in the sense that it yields consistent results with various versions of input data.  I will put this claim to the test.

My recent efforts have been along of three main branches.

1. Data format consistency

2. Smoothing accuracy

3. Presentation of results according the VR2009 model

Data format consistency

I have built a library of temperature, sea-level and sea-level modifier (reservoir storage, groundwater depletion, etc.)  data from various sources.  In some cases the data has been from primary sources.  In other cases getting the data has required other techniques, such as digitizing graphs from journal articles.  In some cases there are multiple sources for ostensibly the same data.  I will make all items in this data library easily available online.  I will appreciate and consider suggestions for additions to this library.

All data sets will be saved in the same simple consistent ASCII file format.  This format consists of three sections, “source,” “notes” and “data.”  Each section is designated by a token that my code can distinguish.  The files should easily open in any text reading application or spreadsheet.

The advantage of this format is that I can use the same code to access data of any type.

The “source” section tells where I acquired the data.  For example, it may list the URL for an online source, or journal source information.

The “notes” section is a catch-all that may mention various things like a particular figure or image in a journal (for example, if the data were digitized from a graph), additional processing that I may have applied (for example, the original data might be sea level rise rate data which I integrated to give sea level).   Anything that I feel is necessary to clarify the data is included in this section.

The “data” section consists of two or more columns of tab delimited data, each with a units header.  The first column is always time, in fractional year format.  For temperature data files the second column is always degrees C.  For sea-level and sea-level-modifier files the data is always relative sea level in mm.  Additional columns are included when necessary and their meanings should be clear from their headers and/or the “notes.”

Smoothing accuracy

As Rahmstorf and others have correctly noted, typical temperature and sea level data is quite noisy and requires a smoothing function to be useful in his models.  The primary problem with smoothing functions and curve fitting techniques is their tendency to give inconsistent or erratic results at the ends of data series.

In Rahmstorf’s 2007 science paper (R2007)[3] he says “Both temperature and sea-level curves were smoothed by computing nonlinear trend lines with an embedding period of 15 years.”  Rahmstorf referenced “New Tools for Analyzing Time Series Relationships and Trends”[3] for his nonlinear trend line smoothing technique.  This paper in turn referenced others, etc.   The same or similar techniques were used on his subsequent papers.

Following his references was very much a rabbit hole, but I showed that his results could be very closely reproduced using a much simpler smoothing technique (for example, see here and here).  I used a Gaussian filter with a blended polynomial fit at the ends.

I have spent considerable time examining and testing various smoothing techniques over the last few months.  Some very elaborate, some very simple.  I have settled on a relatively simple method of applying a Gaussian filter to data with linearly extended ends.  In essence, the end regions of the original raw data series are fit to a line, those lines are extrapolated beyond the ends, a Gaussian filter is applied to the extrapolated series, and finally the extrapolated parts are truncated back off the smoothed data.  This method works consistently very well for all of my data sets.

Presentation of results according the VR2009 model

The data format consistency and smoothing method makes it easy to input any combination of temperature data, sea level data, and sea-level modifiers into my implementation of the VR2009 model.  The output of any combination will be presented in two new files: a simple ASCII file that can be viewed in any text reader or spreadsheet and a jpeg that shows graphical presentations of input and output data.

The output ASCII file will list the sources of all input data, columns of unsmoothed and smoothed input data.  It will include setup parameters like the end linear extension lengths and the Gaussian filter FWHM.  Finally, the  key fit parameters: a, b, and To, and columns of resulting modeled time series for H and dH/dt and the residuals for H and dH/dt will be listed.  All time series data will be tab delimited for easy spreadsheet plotting so anybody can test the quality of my smoothing and fitting.

The jpeg image will show the same items as the ASCII file will all time series data graphical format.

In the works…

I will also add an extension to my code that will apply the results of the VR2009 model with any desired temperature and sea level inputs to temperature projections for the 21st century.  This will yield a range of sea level rise projections for the 21st century.

In the next few days…

I will apply the new code, with the new smoothing techniques to the same sea level and temperature data used in VR2009 and present the imput and output files to compare to VR2009’s results and my earlier reproduction of the VR2009 results.


1.  Rahmstorf, S., Perrette, M., and Vermeer, M., “Testing the robustness of semi-empirical sea level projections” Climate Dynamics, 2011

2. Vermeer, M., Rahmstorf, S., “Global sea level linked to global temperature,” PNAS, 2009

3. Moore, et, al.,  “New Tools for Analyzing Time Series Relationships and Trends” Eos, 86, 2005


  1. There was a lot of blogosphere discussion of Rahmstorf’s smoothing methods back in late June/July/August 2009 after the release of the Copenhagen Synthesis Report.

    It seems that the “nonlinear embedded smoothing” is in reality essentially nothing more than a filter with triangular filter coefficients. The number of coefficients is (2*M)-1, where M is Rahmstorf’s embedded dimension. See http://climateaudit.org/2009/07/03/the-secret-of-the-rahmstorf-non-linear-trend/ for a comparison of how closely this triangular filter mimics Rahmstorf’s results.

    In addition, Steve McIntyre has reverse engineered Rahmstorf’s method and has R code for his filter technique.

  2. Very nice scientific work going all the way back to 2010. My complements. I have rewritten a paragraph in the chapter on climate change to note your critique, with specific attribution to ClimateSanity. Well done.
    Book titled Arts of Truth should be out late fall.

  3. It’s hard to find experienced people for this subject, but you seem like you know what you’re talking about!

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