Lightsum and Darksum are Calculated, not Measured

In last year's post The Tiljander Data Series: Data and Graphs, I explained that the four Tiljander data series were actually three: Darksum is calculated as (Thickness minus Lightsum).

I've since discovered that there are actually two Tiljander data series rather than four.

Thickness and XRD are measured values.

Lightsum and Darksum are values that Tiljander et al. calculated by multiplying Thickness and XRD.

Here are the formulas. Varve thicknesses are measured in microns (thousandths of a millimeter, um).

Lightsum = Thickness * XRD * 0.003937

Darksum = Thickness * ( 1 - ( XRD * 0.003937 ))

Solving these two equations for Thickness yields

Thickness = Lightsum + Darksum

The calculated values of Lightsum are within 0.01% of the values archived at NCDC. For Darksum, the calculated values are consistently 0.5% to 0.8% too low. Presumably, this is a rounding error.

[UPDATE Aug 15, 2011 -- Commenter HaroldW figured out the exact formulas by which Lightsum and Darksum are calculated. It strongly suggests that Tiljander et al. made a minor arithmetic error in their formulae, such that

Thickness = Lightsum + (( 255/254 ) * Darksum )

"Exact" means that the calculated values of LS and DS agree with the archived values to within 0.001%. I've updated the Excel file at BitBucket to reflect HaroldW's insight.]

"Discovered" as used above is tongue-in-cheek. Obviously, the authors of Tiljander03 have known from the outset that this was their procedure. However, this finding is new to me. Presumably, it is also news to the authors of Mann08, Mann09, Kaufman09, and to other people who take an interest in paleoclimate reconstructions.

"Does it matter?" From a statistical point of view, yes, it does.

Before going into why it's important, here are comparisons of Downloaded versus Calculated Lightsum and Darksum. Clearly, the Calculated series are essentially identical to the series as they were archived. (Graphed data available for download as Excel file Tiljander03-Calculated_LS+DS.xls at BitBucket.) The two graphs are followed by a description of the methods used by Mia Tiljander and her colleagues.

Tiljander et al.'s methods led to the measurement of Thickness and XRD. In their fieldwork, they recovered drill cores of thousands of years of varved sediments from the bottom of Lake Korttajarvi. Back in the lab, they stabilized and preserved these cores by a set of procedures that are commonly used for biological specimens. First they infused the waterlogged mud with acetone. Once all the water was removed, they then impregnated the mud with liquid epoxy, which hardened as it cured. Once solidified, the cores could be cut with a bandsaw to yield a specimen with the desired 2-millimeter thickness. (Reference: Tiljander, Ojalaa, Saarinena, & Snowball, 2002; abstract.)

The first analytical step was to determine the thickness of each annual varve, likely with a caliper.

Second, the epoxied core segments were placed atop X-Ray film, and a pre-determined burst of X-Rays illuminated the specimen. As with dental X-Rays, some materials absorb more X-Rays than do others, exposing the film less, or more. After development, the X-Ray film was scanned and digitized. Minerals (e.g. silica) absorb more X-Rays, leaving the underlying film relatively less exposed. Organic matter absorb less of the X-Ray energy, causing the underlying film to be relatively more exposed. Thus, high XRD is indicative of a high proportion of mineral matter; low XRD indicates a high proportion of organic matter.

The final step in calculating how much mineral matter and how much organic matter was deposited at the bottom of Lake Korttajarvi each year is to combine the Thickness and XRD information. The thicker the varve, the greater the total of Mineral matter plus Organic matter. The less-exposed the X-Ray film underlying a varve, the higher that varve's X-Ray Density, and the higher its proportion of Mineral matter. (And, the lower its proportion of Organic matter.)

The equations for Lightsum and Darksum near the top of this post represent these relationships quantitatively.

These procedures are well-described in publications by co-authors of Tiljander03. For example, on page 20 of his 2001 PhD dissertation (1 MB PDF), Antti E.K. Ojala wrote:

our general procedure (Papers III; IV; V) has been to digitise X-ray radiographs with 1000 dpi optical resolution, providing an average of approximately 24 grey-scale data points per one 0.6 mm thick varve... The acquisition of comparable high-quality grey-scale images of the varved section is usually the most critical and time-consuming phase in digital image analysis. Owing to the considerable density difference between a minerogenic spring lamina and organic matter deposited during the summer, autumn and winter, X-ray radiography is an important and useful tool in documenting thinly (< 1 mm) laminated clastic-organic varves. Dense minerogenic layers have a greater ability to absorb X-rays than organic layers, therefore showing a lighter shadow in the X-ray film (Fig. 3). By using a 19-step standard glass sample with known density (Bresson & Moran, 1998), the comparability of the X-ray radiographs (grey-scale) of 2 mm thick slabs of embedded sediment was facilitated (Paper III).

So: why does it matter that two of the Lake Korttajarvi data series are calculated from the measured values of the other two?

The answer lies in the idea of Degrees of Freedom.

From Wikipedia's entry, here is one definition of the concept:

A common way to think of degrees of freedom is as the number of independent pieces of information available to estimate another piece of information. More concretely, the number of degrees of freedom is the number of independent observations in a sample of data that are available to estimate a parameter of the population from which that sample is drawn.

Think of it this way: suppose I wished to use a set of proxies to estimate a time series of something. That 'something' could be anything: temperature, precipitation, or kangaroo population, for instance. Since my proxies are noisy, I'll have more confidence in an estimate that is derived from a larger number of proxies -- all things being equal. But suppose I decided to increase the proxy count by copying-and-pasting columns in an Excel spreadsheet. One proxy can become two! Two can become four!

Obviously, this sort of copy-paste activity can't improve my results, because I haven't increased the number of independent observations in the data I am using to estimate the parameter of interest (temperature/precipitation/kangaroos). In other words: bigger spreadsheet, but unchanged degrees of freedom.

Returning to Tiljander-in-Mann08:

If Lightsum and Darksum are used as "proxies," then Thickness and XRD cannot be used without specifically reducing the d.f. in all calculations -- they aren't independent.

Conversely, if Thickness and XRD are used as "proxies," then Lightsum and Darksum cannot be used without specifically reducing the d.f. in all calculations -- same reasoning.

There are two possible results if these cautions are not observed. First, the Tiljander data series will be overweighted -- there seems to be twice as much independent data from Lake Korttajarvi as is actually the case. Second, confidence intervals will be drawn too narrowly, as degrees of freedom always enter into such calculations. How much overweighting? Since none of the Tiljander data series can be directly calibrated to the instrumental temperature record -- Mann08's sole approach -- that question can't be answered. (As discussed in other posts, the proper weighting of the Tiljander data series is "zero".) How much underestimation of confidence intervals? There doesn't seem to be a clear answer to this question, either, for the same reason.

In my opinion, there's no evidence and no likelihood of any intent to cut statistical corners by the authors of Mann08 or Mann09. The simple and obvious explanation is inadequate due diligence. This appears to be a common shortcoming of the "proxyhopper" approach favored by these and other researchers engaged in paleotemperature reconstruction.