# Climatic Cycles and Tree-Growth/Chapter 5

​Result of study of curves. — On completing numerous curves of tree-growth in the manner already described, three characteristics were observed: (1) in arid-climate groups the annual rings are approximately proportional to rainfall; (2) in moist-climate groups they vary with the changes of solar activity; (3) in each they are subject to certain cycles or periodic variation. The first of these is the subject of the present chapter.
Early tests of rainfall correlation. — The earliest comparison with rainfall in this investigation was made between the first Flagstaff subgroup of 6 trees and 43 years of precipitation records at Prescott, 67 miles distant. It was not expected that agreement in individual years would be found; accordingly smoothed curves were used, consisting of overlapping means of 9-year groups. This produced curves of gentle variation, but similarity in the curves was evident. These early curves are presented in figure 13. The best agreement was found
Fig. 13.—Correlation between tree-growth and rainfall in smoothed curves; Flagstaff.
by placing each mean of 9 years of rainfall at the end of the 9 years as in this figure instead of in its center. This lag of four years seemed inconsistent with the later results of yearly agreement without lag, and in fact for years it has been accepted with some hesitation by the writer. Yet in the present consideration of the subject it appears to have a special significance. This existence of the lag in long periods agrees in principle with the "accumulated moisture" effects observed in the Prescott trees and with the idea of a tree exhibiting a reserve power or vitality which may run low or be built up by varying environment. The principle will be referred to again below; it is sufficient now to state that it seems quite reasonable to find no lag in yearly correlation with rainfall and at the same time a very considerable lag in the slower variations.
The comparison in figure 13 was made with Prescott records because there were not at that time enough Flagstaff records to be of service. But later, when a Weather Bureau station had been established in Flagstaff for several years, the striking comparison shown in figure 14 ​ Fig. 14.—Early test of correlation between tree-growth and rainfall by years; Flagstaff.
was made. In this the lower curve represents the average annual growth of 25 trees and the upper curve is the precipitation 12 miles distant. The latter is taken from November 1 to November 1 in order to carry the snowfall into the following season of growth. This study suggested the investigation of the time of year to begin annual means of rainfall, which has already been presented in Chapter II. Figure 4 gives a comparison between Flagstaff rain and the two Flagstaff groups, and also shows how the best time of beginning the year was determined. It proved to be November 1 at Flagstaff and September 1 at Prescott, where the nature of the ground gives more chance of conserving moisture. The great difference between individual trees in response to rain is also shown in figure 5. It is evident that quick-growing trees serve as better indicators.
THE PRESCOTT CORRELATION.
Five subgroups, numbering in all 67 trees, were obtained from different points in the vicinity of Prescott. These all cross-identified among themselves with entire success, both as individuals and as groups. The group curves are shown in figures 6 and 7, but in comparison with the Prescott rainfall they differed greatly, the group nearest the city showing much the best accordance. Accordingly this group is plotted by itself in figures 7 and 15 with the rainfall curve. On the whole there is much agreement, as may be seen by comparing the crests and troughs of one with those of the other. The most conspicuous discrepancy is in 1886, where the rainfall decreases and the growth of the trees increases. In 1873 the growth seems to have responded to the decrease in rainfall, but to a greatly diminished degree. The tree maximum of 1875, one year behind the extreme maximum of 1874 in the rainfall, is entirely reasonable, since the ground may become so saturated that the effects last until the following year. On the whole, the curves shown in figure 7 support the idea of a proportional relation between annual rainfall and annual growth.
Accuracy.—The accuracy with which the pine trees near Prescott represent the rainfall recorded in that city for 43 years is, without correction, about 70 per cent. By a provisional correction for conservation of moisture by the soil this accuracy rises to about 82 per cent. The nature of this conservation correction is very simple; it makes use of the "accumulated moisture" of the meteorologist. It signifies that the rings in these dry-climate trees vary not merely in proportion to the rainfall of the year, but also in proportion to the sum ​of the profits and losses of the preceding years. The "credit balance" in their books at the beginning of the year has only somewhat less importance than the income during the current year.
Mathematical relation of rainfall and growth. — In order to formulate the relation between rainfall and tree-growth, an effort was made to construct a mathematical formula for calculating the annual growth of trees when the rainfall is known. Any such formula must perform three principal functions: (1) reduce the mean rainfall to the mean tree-growth; (2) provide a correction to offset the decreasing growth with increasing age of the tree; and (3) express the degree of conservation by which the rain of any one year has an influence for several years. In a formula of universal application other factors will play a part, but for a limited group of trees in one locality they can be neglected.
The first process, namely, the reduction of the mean rainfall to the mean tree-growth, is a division by 250. This is the general factor K in the formula below. The second part, namely, the correction for the age of the tree, was practically omitted in forming the curves shown, since judging by the Flagstaff curves its effect would be very slight in the interval under discussion. In long periods it is an immensely important correction and its effect should always be investigated. Over the short periods used in this rainfall discussion the decrease of annual growth with age may be regarded as linear and an approximate formula is
${\frac {G_{n}}{G_{y}}}=1-c(n-y)$
Where G$_n$ represents growth in any year n; G$_y$ is growth in middle year of series y, and c is the rate of change per year, a constant which was 0.0043 in the last half century of the Flagstaff series. Over the whole interval from 1700 to 1900, in the first Flagstaff curve, the growth was approximately an inverse proportion to the square root of the time elapsed since the year 1690 and may be closely expressed in millimeters by the formula
${T_{n}}={\frac {10}{\sqrt {n-1690}}}$
T$_n$ is here the mean tree-growth for the year n. If G be the mean size of rings, then the factor to be introduced in a general formula becomes
${\frac {10}{G{\sqrt {n-1690}}}}$
Character of the conservation term. — This factor has two important features: (1) in this arid climate it applies better as a coefficient than as an additive term, and (2) it gives a prominent place to "accumulated moisture" as commonly used in meteorology. ​The first assumption in regard to conservation was that the ring-growth in any one year was built up by contributions from the current year and previous years in diminishing proportion. For example, it would be proportional to
$R_{n}+{\tfrac {1}{2}}R_{n-1}+{\tfrac {1}{4}}R_{n-2}$ etc.
in which R$_n$ is the rainfall for the current year, R$_{n-1}$ that for the year preceding, etc. This may be called an additive correction. It did not
Fig. 15.—Relation of tree-growth and rainfall at Prescott, Arizona. Tree-growth and rainfall uncorrected.
Fig. 16.—Five-year smoothed curves of growth and rainfall.
Fig. 17.—Accumulated rain and smoothed tree-growth.
Fig. 18.—Actual tree-growth and growth calculated from rain.
Fig. 19.—Actual rain and rain calculated from tree-growth.
give satisfactory results for the Prescott trees, although a formula of this general type has been applied with some success to the sequoia, which grows in more moist soil.
The variations in the Prescott trees were seen to be proportional both to the rainfall of the year and to the average growth or activity ​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. ​3. From A$_n$ an approximate R$_n$ is found by the formula
$R_{n}=M+A_{n}-A_{n-1}$
4. This series of approximate rainfall R$_n$ is smoothed and becomes the S$_n$ of the formula.
5. Final values are then found by the proportion
${\frac {0.90M+{\tfrac {1}{4}}A_{n}}{250}}:S_{n}::T_{n}:R_{n}$
It should be emphasized that the above formula for conservation is the one found to apply under dry climatic conditions. In moist climates the trees, so far as observed, seem to depend on other meteorological elements or combinations of elements.
The Prescott trees, as we have seen, even without correction give a record of rainfall with an accuracy of about 70 per cent. It is possible that the Flagstaff trees with their higher elevation, more certain rainfall, and more central location in the zone occupied by this species give somewhat more accurate records. They are probably much less often subjected to extremes of dryness, which throw the tree out of its equilibrium and cause it to produce an abnormally small set of rings. It seems likely, also, that the less compact soil, combined with a more abundant precipitation, produces a yearly growth more nearly proportional to the rainfall than at Prescott.
Summary. — In considering this reduction it seems fairly clear that (1) there is a strong correlation between rainfall and tree-growth; (2) the accuracy may be increased by introducing a conservation correction; (3) in dry soils this factor enters as a coefficient; (4) this coefficient depends on the state of activity of the tree; (5) in the Prescott trees this state of activity follows the curve of accumulated moisture.
Although the moisture-content of apparently dry ground may be a most important item, it is by no means certain that the observed accumulated moisture effects consist in actual moisture in the ground. It may be that they represent some vital condition of the tree. The matter is a very interesting one for future study.
Sequoia correlation with rainfall. — On his return from the big trees in 1912, Professor Huntington supplied me with a curve of sequoia growth obtained from many comparatively young trees which had been cut in the lower edge of Redwood Basin near Camp 6. On comparing these with his curve of rainfall in the San Joaquin Valley, compiled from records at Fresno and San Francisco, a close relation was not evident, but an additive formula
$T_{n}=k\cdot {\frac {R_{n}^{2}+R_{n-1}^{2}+R_{n-2}^{2}}{R_{n}+R_{n-1}+R_{n-2}}}$
was used with encouraging results. This formula was designed to allow for strong conservation in the soil, not of the static type as in a ​pond, but of the moving type, as if a belated supply from the snows came to hand and then passed on. The tree was assumed to receive moisture from the current year and from the first and second preceding years; and whichever of the three was greater, that one had the more effect. The application of this formula is shown in figure 20.
Fig. 20.—Huntington's early curves of sequoia-growth and rainfall compared with growth calculated by a conservation formula.
But on identifying the rings in the trees collected from that locality in 1915, and especially on finding the soft, delicate parts of the 1915 ring on D-5, it seemed fairly certain that the curve of growth given in figure 20 is one year in error through the omission of a final ring. The growth, then, which appeared to be 1902, for example, and for which a pronounced conservation was necessary, really came the year before,
Fig. 21.—Comparison of Fresno rainfall (after Huntington) and sequoias D-1 to 5.
and less conservation or none was needed. The comparison of the same rainfall curve with the old sequoias of the present series is given in figure 21. In this the agreement is not as good as at Prescott, but there is marked similarity in many details. My curve from very old ​trees is probably not as good in details as Huntington's samples from young and sensitive trees. His material is well worth cross-identifying and dating with care, and then comparing with any records of snowfall which can be obtained from the sequoia groves. It is greatly to be regretted that Fresno, 65 miles away and at 5,000 feet lower elevation, is the nearest point where precipitation records can be obtained for a period long enough to be of value.
Future work.— It will be very interesting to find whether the characteristics of the correlation at Prescott are general in arid climates and dry soils and whether practical formulas for conservation in moist soils or climates can be worked out. When this is done the significance of the study of annual rings will be greatly increased.
METEOROLOGICAL DISTRICTS.
The study above described raised emphatically the question as to the extent of the region or district from which comparative rain records should be selected. Such a meteorological district could be defined as one in which homogeneous weather elements are found. But we immediately ask ourselves the questions: must all weather elements be alike in it or is it sufficient to have only rainfall (for example) essentially the same throughout; will the district remain constant through indefinite time or will it change; is the district for short-period weather changes the same as the district covered by secular changes. In the present discussion I have understood by meteorological districts such regions as may show similar or identical variations in some one weather element. It seems likely that a region which may show unity in small or rapid variations may not do so in large and slow variations, or more likely may be a small fraction of a region which will show unity in large variations.
Meteorological districts and growth of trees. — The cross-identification of trees over large areas has already suggested the use of annual rings as a possible aid in delineating meteorological districts. This function of the rings has received some exemplification in the present study. For instance, the pine trees of Norway differed in such a way that it was necessary to divide them into two classes, one of which came from the outer coast near sea-level and the other from the inner fjords and mountains. The trees from these different regions show strong reversal with reference to each other. Again, the trees from the lowlands about the Baltic Sea show marked similarity in their variations and indicate, as we would expect, a homogeneous district. Furthermore, groups from near the Alps show strong differences from the other European groups, as we might expect from our experience with the five groups from the mountainous country about Prescott. A rugged and mountainous region is very difficult to divide satisfactorily ​into meteorological districts. Yet, in spite of local differences, mountain regions may be alike in major characteristics, for all the Prescott groups, though differing among themselves, cross-identify excellently with the Flagstaff trees 60 miles away. The sequoias also cross-identify perfectly in mountain localities 50 miles apart, showing that there is enough similarity in different parts of the high Sierras to cause the trees to agree in many variations.
Arizona and California. — Fully 450 miles intervene between the sequoias of California and the pines of Arizona, yet there are strong points of identity between them in the last 300 years. The dates of notably small rings are much alike in each. The details of this comparison have not yet been fully studied, but they support the idea long since expressed (1909) that Arizona and California, especially its southern half, form parts of a large district which has similarity in certain variations. A long acquaintance with this region throws light on the details of this similarity. The winter precipitation, which is largely in the form of snow at the altitude of the trees studied, has the major influence on tree-growth, for it is largely conserved near the trees, whereas the summer rains are usually torrential and the water quickly flows away. The winter storms moving in an easterly direction reach the coast region first and after about 24 hours are felt in Arizona. Thus, in spite of the coast range of mountains and the intervening low-level deserts, each winter storm passes over both regions and causes an evident similarity between them. In a large view they belong to a single meteorological district.
Meteorological districts and solar correlation. — In searching for a link of connection between solar variation and meteorological changes, we must bear in mind the effect of possible reversals in neighboring meteorological districts, such as noted above in Norway. It may be the lack of such precaution which has caused many meteorologists to condemn at once the suggested connection between the distant cause and the nearby effect. We must remember that districts may be small in area, and in combining many together we may neutralize the result for which we are in search. Some illustration of correlation found in small districts will be given in the final chapter.