Dust flux, Vostok ice core

Dust flux, Vostok ice core
Two dimensional phase space reconstruction of dust flux from the Vostok core over the period 186-4 ka using the time derivative method. Dust flux on the x-axis, rate of change is on the y-axis. From Gipp (2001).

Tuesday, July 20, 2010

First steps into complexity part 1

I will try to document some of my thinking as I moved from a standard mechanistic viewpoint of science to one that was more complex.

I have been involved in Quaternary climate studies since I began my MSc in marine geology at Memorial University of Newfoundland. There I worked with Dr. Ali Aksu ostensibly on a typical marine geology study of a sedimentary basin on the continental shelf of Nova Scotia, but I also spent some time pondering Quaternary climate change--in particular, the Milankovitch theory of astronomically driven climate change.

At first the problem was a straightforward technical problem--how to tease out the appropriate signals from marine records. In the course of background reading, I encountered a relatively unknown paper (at least by geologists), by E. N. Lorenz in Quaternary Research in 1976 (far more famous were his earlier works on nondeterminism in weather prediction). The QR paper presented alternative ideas concerning the fundamental architecture of the global climate system and challenged the geological community to test them and so determine the nature of climate change on the Quaternary timescale. Most of the literature of the time considered the climate system to be deteministic, while yet acknowledging that there were nonlinearities which complicated the whole thing--but it was clear that the nonlinearities were hoped to be local in nature and that they could be dealt with through a judicious series of fudge factors. Lorenz described three possibilities for climate: 1) a straightforward "transitive" system, in which the system outputs can be linked to the system inputs by a simple set of differential equations; 2) what he called intransitive (what we would now term multistability; i.e., a system as above but with different sets of differential equations operating at different times); and 3) what he termed "almost intransitive", and called "strange attractors" in other publications, and which we now refer to as simple chaos.

I say that the paper is poorly known as I have never seen any commentary on it. Nor, for many years, did there appear to be a clear attempt to distinguish among these different modes of operation. To be sure, there have been publications advocating any one of these modes (here and here), but most of these were attempts to show observations which supported the proposed mode, rather than using observations to test between the different modes. More recently, various climate models have been proposed in which the modal operation is taken as a given.

I finished my MSc., then shifted to University of Toronto to carry out Ph.D. research with Dr. Nick Eyles. My principal thesis was again a geological one which concerned itself with tectonic influence on the development of glaciated continental margins, using Eastern Canada and the Gulf of Alaska as contrasting examples. However I also devoted a lot of time to Lorenz's proposed problem of Quaternary climate change. My approaches to this problem followed several branches. The first was improved signal processing (mainly through alterations in Fourier transform, including attempts to use maximum entropy or other methods). The second was looking at other data sets. The third involved developing entirely new techniques for processing information.

This last approach very quickly came to absorb most of my spare time.

In 1990, the concept of fractals had been around for awhile, but its application in earth sciences was still very much leading edge (I was actually thinking of the first edition of this book). The push to educate earth science professionals had only just begun. At Scarborough there was a post-doc in geography who was trying to make a name for himself by publishing paper after paper in which he reported the fractal dimension of some geographical feature. He had published something like a dozen papers in a year, each of which I must assume, was very short.

The concept of nonlinear dynamics was also very cutting edge in earth sciences. I proposed teaching a course on the topic, going so far as to propose that we teach our own mathematics to earth science students, but the idea didn't go anywhere.

I had encountered an interesting idea in a paper by Imbrie and Imbrie, in which they proposed that it was not ice volume directly that responded to solar insolation, but the rate of change of ice volume. At the time this struck as me as a brilliant insight, and I immediately constructed a figure showing the connection between insolation in the northern hemisphere and the rate of change of ice volume calculated from first differences from a deep sea O-18 record.

Plot comparing insolation at 65N and the rate of change of ice volume from a deep sea O-18 isotope record. A panel from the ill-fated Paleoceanography paper described below.

I then had the idea of constructing a figure in which I plotted the inferred global ice volume against its rate of change, once again calculated from first differences. The graph would be a curve, in which each point would represent the "state" of the system at a particular time, and when all the points were plotted in sequence, a trajectory would be traced which should reflect the dynamics of the ice volume system.

Part of my first two-dimensional phase space reconstruction of global ice volume. The small numbers represent time in thousands of years before present (ka).

Points on the graph that lie above the x-axis represent intervals where ice volume is increasing, and ice volume is decreasing over the segments of the trajectory below the x-axis. The further from the x-axis, the more rapid the growth or retreat of global ice.  The plot above shows the relatively slow advance of glaciers from the period beginning about 120 ky ago until about 20 ky ago, followed by rapid deglaciation.

What was immediately noticeable in observing the function over the past 500 thousand years was that there were particular areas on the graph to which the function seemed attracted. It moved very rapidly towards them, and tended to stay in them for long periods of time before rapidly moving to another. All of these regions plotted along the x-axis, and corresponded to particular volumes of global ice. The location made sense, because it implied that there were particular volumes of ice which were more stable than others. During the times when ice volume was stable, its rate of change must be low--hence it would be impossible to find a small region of attraction off the x-axis.

Now this is a phase space portrait, in two dimensions, using the time-derivative method (Packard et al., 1980). At the time I did this, I didn't know what to call it. I was certain it had been done before, but even in today's world of search engines, without knowing the terminology it is very difficult to find information. I knew that I was on to something, but didn't know what.

In the meantime, I had had another idea for testing climate records for multistability--at least this was a test to distinguish multistability from the transitive case using information theory (I didn't understand enough about simple chaos to devise a test for it). My approach was that if climate had one or more stable states, then there should be measurable differences in the information between the climate record (again the deep ocean O-18 isotopic record) and the driver (which was presumed to be northern hemisphere insolation). If there were multiple stable modes of climate, then the insolation would be encrypted, as if by a polyalphabetic key, and there would be a change in a particular quantity called the index of coincidenc, which is the likelihood that two randomly selected characters in a string of text are identical. There were challenges in applying this, not the least of which that it required that the data should be 'binned' and it was not at all clear how the bin size in the observed data stream should be linked to that of the northern hemisphere insolation. This work was presented at two conferences in 1991 and 1992, and was awarded a top student paper prize in 1991. But when I wrote the paper and submitted it to Paleoceanography, I overlooked one of the cardinal rules of scientific writing.

Always look like you know what you are doing.

I have always been fascinated by the intellectual process of the scientific endeavour. This fascination lead me to make a basic mistake in presenting my experiment and results. In the course of my work I had discovered what appeared to be a novel use for the process of autoencryption--by which I mean using the message as its own key in a polyalphabetic substitution cipher. The charming result is a coded stream that cannot be unambiguously decrypted even by an intended recipient who has been furnished with the key. Such a method of encryption, understandably, had no real application, and so the behaviour of the index of coincidence for this style of encryption was not well known. However I did not discover this until I was forced to come up with an explanation for a rise in the index of coincidence in the observed signals (compared with the presumed driver).

So I wrote the paper this way. Testable hypothesis with two possible outcomes, conduct experiment, find unanticipated outcome, explain why unanticipated outcome was left out of the original hypothesis, modify hypothesis, conclusion. The paper was rejected. It may have been accepted had I submitted the modified hypothesis as the original one, tested it, and reported a result. I had thought that the process of discovery would be interesting to others. In the case of peer-reviewed journals, this view was mistaken. In the course of revisions, I came to realize that the binning issues mentioned above were unresolvable, and reluctantly abandoned this approach, returning to the reconstructed phase space portrait.

No comments:

Post a Comment