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.