1 Carbon Accounting and Averaging Michael L. Roderick, Stephen H. Roxburgh & Belinda Barnes CRC for Greenhouse Accounting Research School of Biological Sciences The Australian National University 21 December 2004 Background “Scaling up” is a commonly used phrase in carbon accounting and throughout the ecological sciences. By the term, people usually mean that measurements are made at local scales (e.g. leaves, plants), but estimates are required at larger scales, e.g. forests stands, continents, global. This disparity between what is (traditionally) measured and what is needed (i.e. stand to continental scale estimates) is often referred to as the “scaling problem”. There is little doubt that there are significant practical difficulties in making suitable measurements; however, there are also some theoretical considerations. This short note describes one of the significant problems – the problem of averaging in the process of scaling up. Mathematical formulations typically incorporate products and quotients of system variables, and the problem of averaging occurs when averages of these terms are taken. The basis of the problem is that the average of a product (what we usually want) is not necessarily the same as the product of the averages (what is often calculated), and the difference between the two may be significant. Similarly, for quotients, the average of a quotient is not necessarily equal to the quotient of the averages. In this note we provide an example, based on a fundamental task in terrestrial carbon accounting - estimating the carbon storage in forest trees, to illustrate the importance of this issue. The Problem of Averages in Carbon Accounting Assume that for a stand of trees we require an estimate of the total carbon. For each individual tree, we can express the carbon (C i , kg) as, i i i d di i i i d i d i i V V m C V V m m C C , , , = = (1) where (kg) is the dry mass, V (m i d m , i 3 ) is the volume and C di (= i d i m C , ) is the mass fraction of the dry matter which is carbon, for the individual. In practical applications C di is usually set at a constant value of about 0.5. (Note that it would be 0.44 if all the dry matter in a tree was cellulose.) The ratio i i d V , ] i D m is known as the basic density in the forestry literature and denoted here as[ , so that, from above, we have, i i di i V D C C ] [ = . (2)