Page:Climatic Cycles and Tree-Growth - 1919.djvu/87

which the trees had exhibited in the preceding few years. But this average growth bore the same relation to the average or smoothed rainfall that the accumulated moisture bore to the smoothed rainfall. Hence the ratio between accumulated moisture and smoothed rainfall gave at once the required ratio between smoothed tree-growth and smoothed rainfall. These relations are shown in figures 16 and 17.

Accumulated moisture is simply the algebraic sum of the amounts by which all the years in a series from the start to and including the year desired depart from the mean. It may be expressed by a formula, thus

$A_{n}=(R_{1}-M)+(R_{2}-M)+....(R_{n}-M)=R_{1}+R_{2}+R_{3}+...R_{n}-nM$

and conversely

$R_{n}=M+A_{n}-A_{n-1}$

In this formula A_n is the accumluated moisture for the nth year of a series of consecutive years whose mean rainfall is M.

The simple empirical formula for the tree-growth T_n for the nth year of this series thus was found to be:

$T_{n}=K\cdot {\frac {cM+dA_{n}}{S_{n}}}\cdot {R_{n}}$

in which c and d are small constants found advantageous in reducing the accumulated moisture curve to proper scale. S_n is the reading of the smoothed rainfall curve and the term $cM+dA_{n}$ is the accumulated moisture expressed in values above a base-line instead of departures from a mean. In actual numbers this becomes

$T_{n}$(in inches)$={\frac {1}{250}}\cdot {\frac {0.90M+{\tfrac {1}{4}}A_{n}}{S_{n}}}\cdot {R_{n}}$(in inches)

The mean value of the rainfall M is 17.1 inches. The application of this formula in calculating tree-growth at Prescott from the rainfall is shown in figure 18.

The reversal of the process in order to ascertain rainfall from tree-growth seems to be fully as accurate over this limited period, and its result is shown in figure 19, where the curve has an average accuracy of 82 per cent for individual years. In producing this reversal the following operations were performed:

1. A 5-year smoothed curve was made of the tree-growth. This gives us the term ${\frac {0.90M+{\tfrac {1}{4}}A_{n}}{250}}$ in the reversed formula

$R_{n}={\frac {S_{n}}{\frac {0.90M+{\frac {1}{4}}A_{n}}{250}}}\cdot {T_{n}}$

2. This term is multiplied by 1,000 and 3.6M subtracted, leaving A_n in inches.