IV. DETAILS OF CURVE PRODUCTION.
PREPARATION OF RADIAL SAMPLES.
Form of sample.—Nearly all of the 230 trees used in this investigation are represented by portions preserved in my collection. Wherever possible the entire section, 1 to 3 inches or more in thickness, was brought to the laboratory for examination. Unless the section was light and easily handled, it was found convenient to cut from it a radial piece showing the complete series of rings from center to bark. Naturally the enormous trees of the sequoia groups could be obtained only in radial form. The paper rubbings from Oregon and the small cuttings of the Prescott and second Flagstaff groups were also of this type. Hence the radial sample is regarded as the usual or type form in which the material appears in the laboratory. If the original section was small the radial piece appears as a bit of wood cut across the grain, square or triangular in cross-section and a foot, more or less, in length.
Method of Cutting.—The partial radials, such as used in the Prescott group, were secured from the stumps in place by making saw cuts at the edge of the stump in two directions, meeting a few inches below the surface. In this manner a piece of wood in the form of a triangular pyramid was secured and was sent to the laboratory. The radials of the sequoias were cut altogether from the tops of stumps or from the ends of logs that lay on the ground. From the manner in which the trees were cut down it was usually possible to get a clear surface of stump or log from the bark on one side to somewhat past the center where the under-cut had been made. After a minute examination of the surface exposed, a radius was selected which would give the greatest freedom from fire-scars and other irregularities of ring distribution. Two lines about 8 inches apart were drawn with blue chalk along this radius. Then two men with a saw 8 to 14 feet in length made a slanting cut on one of the lines of sufficient depth and in the right direction to meet a similar slanting cut from the other chalk line. In this way a long piece of wood of V-shape in cross-section was obtained, extending from the center to the outside and giving the full ring record.
In sequoias recently felled this cutting of the radials was extremely easy, but many of the sections obtained were from stumps which had been standing and weathering for 25 years and in one case 43 years. The exposure carbonizes the top of the stump and makes it extremely brittle and difficult to cut; small pieces break off and wedge the saw. Thus it often becomes a very difficult task to extract the radial section. The pieces into which the radial section breaks are marked for identification immediately, photographed and listed in notebook, and then carefully packed for shipment. On arriving at the laboratory, they are pieced together with the greatest care and then glued together in groups, making the entire radial section a series of convenient pieces about 2 to 3 feet in length. Preparation for measurement.—These pieces were then examined to find the longest sequences of clear and large rings, and guide-lines for the subsequent identification and measurement were selected as nearly as possible perpendicular to the rings. Such lines having been decided on, two straight pencil lines, half an inch apart, were drawn and the surface between these was "shaved." For this purpose, after the trial of many other methods, a common safety-razor blade was clamped to a short brass handle. With this very sharp blade the rough surface of the wood is removed and the rings stand out very clear and distinct. Besides the space between the lines, the region close outside is usually shaved also for a preliminary trial at cross-identification, the final marks being the only ones permitted between the guide-fines.
The best light for observing the rings is a somewhat diffused light coming sharply from the side. A light falling on the wood perpendicularly is apt to be very poor, either for visual work or photography. Light from each side must be tried, for there is often a great difference between the two directions, due probably to the way in which the knife passed over the wood and bent the ragged edges of the cells. In photographing, the colors involved and the result sought (i. e., to show the red rings as black) require an ordinary plate and a blue color-screen.
When the surface is well prepared it is placed in a suitable light and wet with kerosene applied by means of a bit of cotton on the end of a small stick. This deadens the undesired details of the surface, and brings the rings into greater prominence. The identified section is now supported over the unknown and with watchmaker's glass in eye and long needle in hand, the observer can make rapid comparison and quickly put on the required marks.
IDENTIFICATION OF RINGS.
In the early Flagstaff work the rings were first numbered, beginning at the outside without regard to the year in which they grew. But this was found to add complexity and involve the use of a separate reduction from the provisional numbers to the true dates of the rings. Accordingly the rings are dated at once as well as possible on some selected section that gives promise of an accurate record. The identification mark is a pin-prick or very small hole placed on the last ring of each decade. The middle year of each century has 2 pin-pricks and the centuries are marked with 3; the 1,000-year mark is 4. Marks found in error are "erased" by a scratch through them. After the selected section is dated with the greatest care not to overlook or mistake any rings, others are dated by direct comparison with it. The common practical test in such comparison is the relation of width of a ring to its half-dozen near neighbors. For some unknown reason, rings of diminished size seem to carry more individuality than enlarged rings, and so they are usually picked out for cross-comparison. In nearly every decade some are thus distinguished, and in each century there are usually 3 to 4 conspicuously small rings which give very important aid.
In the first work on the 2,200-year sequoia record, the identification was a laborious task involving all the writer's spare time for a year. The only real difficulty was with the ring for the year 1580. This was temporarily called 1580a, but the material collected in 1919 showed it to represent a year and a final and complete renumbering included it as such. In the end the comparisons gave entire confidence as to the identity of every ring. Section No. 2 gave the most nearly perfect long record, beginning at 274 B. C, and is used as a standard with which to compare all new ones.
The most difficult parts to identify are the compressed rings. Over long periods, varying from 5 or 10 up to 100 years, the rings are sometimes so crowded together that large numbers of them seem to be merged into one and their identification becomes extremely difficult and in a few cases impossible. The great variations in sizes so produced also exaggerate effects. These groups of compressed rings are considered as of little value, and in fact in many trees their measurement is omitted altogether. Tree No. 12 of the sequoias obtained from the Indian Basin had such bad groups of compressed rings that it proved practically impossible to identify them without a large expenditure of time not then available. Tree No. 17, also, from Camp 7, was found so full of compressed rings in the last few hundred years that all measurements were omitted after the year 1130 A. D.
Fire-scars. — Most of the big trees show fire-scars at some time in their history, and the process of the tree's regeneration is very interesting to observe. If the scar is small the woody growth quickly comes in from each side and covers it. If the scar is very large, occupying perhaps one-quarter or one-third of the circumference, the tree is likely never to recover and the burnt place remains permanently on its side. In cases of less extensive burns, the wood from each side year by year grows toward and over the injured spot, and if the injury has not been too great the approaching sides may meet and imprison their own bark within the tree. Thus one often sees the tops of the stumps marked here and there by a hole as large as a foot in diameter, filled with bark in perfectly good condition.
No. 12 had several fire-scars that interfered with the identification of rings. No. 18 also had one or two fire-scars and in particular showed a fire in the year 1781. The latter evidently stopped the growth at that point completely, yet was not large enough to interfere with recovery. In the sample in the laboratory the usual reddish-colored heartwood changes about the year 1700 to the white sapwood, which ends with the ring 1781 and shows a surface that was once covered with bark. However, immediately outside of that surface, the red heartwood begins again with the year 1791 in a thick, rapid growth. The heartwood continues for some 20 years before changing again into the white sapwood, which persists to the outside. In order to make sure that this gap would not prevent satisfactory identification, a small portion was cut from another part of the outside of the tree, showing some 300 rings without interruption; but this additional piece I found in that case to be unnecessary.
In sections numbered 22 and 23, from the old Enterprise millsite, there are injuries which do not greatly alter the appearance of the rings, yet are sufficiently great to weaken the wood and cause it to break at several points. If the break in such case is across the rings, it is easy to carry the identity of rings past the injured point. But when the break in any wood "sample is all in one ring there may be a doubt as to whether the break is between two rings or in the middle of one. In the latter case there will apparently be an extra ring at that point. If the break is obviously between two complete rings, then an unknown number of rings may be lost at the broken point. The only way to carry the correct dating of the rings past such broken places is to secure samples from other parts of the same tree or from other trees, which show 100 to 200 rings on each side of the uncertain place without serious interruption. A simple cross-identification will show whether any rings are lost. However, in Nos. 22 and 23 just referred to, nearly all lines of breakage crossed the rings in a way that left no uncertainty. But No. 22 had an injury and a break between complete rings at about 1020 B. C. and a pronounced injury at about 1060 B. C. No. 23 had an extensive decayed place with the loss of about 35 rings at 1060 B. C. An extra piece cut from the stump of No. 23 carried the dating across these gaps with perfect satisfaction and in complete accord with No. 21 which had been secured 50 miles to the north.
Cross-identification between distant points.—The sequoias collected in 1915 had come from the immediate vicinity of Camp 6, about 7 miles east of Hume, and from Indian Basin, which is 3 to 4 miles north of Hume. The total extent of country covered was about 10 miles. All these were identified and found to be very similar in their characteristics. In 1918 the country represented was extended by sections from the new Camp 7, some 2 miles east of Camp 6. Nos. 20 and 21 were then obtained from the old Converse Hoist, 4 miles from Indian Basin and 15 miles from the Camp 7 district. Finally, 2 trees were obtained from the old Enterprise millsite, 50 miles from the other localities. It was realized at the time that there might be difficulties of cross-identification between these 2 trees at Enterprise and the other well-known and well-identified groups near Hume and the General Grant National Park. However, it was very gratifying to observe on close examination of these sections that no uncertainty was introduced in the identity of the rings. One realizes from this that, so far as sequoias are concerned, a distance of 50 miles between groups is likely to be no particular obstacle in cross-identification.
The difficult ring 1580.—The small ring 699 A. D. and several other difficult ones were absent in comparatively few trees and any uncertainty regarding them was removed in the early part of the work, but it was not so with the ring of the year 1580. The best of the tree records were from the uplands and usually omitted it, while many of the basin trees which showed it were at first very uncertain in identification. The ring was therefore provisionally called 1580a and held in doubt for several years. The question of its reality was finally settled in the affirmative by a special trip to the sequoias in 1919 and the collection of a dozen carefully selected radial samples. The final review of all the tree-records has resulted in satisfactory identification of some previously doubtful cases and in complete conviction regarding the ring for 1580 A. D. No other uncertain cases were discovered. Considering the 35 sequoia records now (1919) made use of, it seems possible that all errors of dating have been removed.
MEASURING.
Having prepared and identified the wood samples, the first method of measuring was to lay a steel rule on edge across the series of rings in a radial direction and to read off from the rule the position of the outside of every red ring. These were either recorded at once by the person measuring or were noted by a clerical assistant. This method applied to the Flagstaff and Prescott trees and to the European and Vermont groups. In nearly all of them the steel rule used was a meter in length. It was ascertained by tests that the errors in readings of this kind were less than 0.1 mm. on the average for a single reading. For the Oregon group a microscope slide was used with a vernier which gave at once readings to 0.01 mm. The readings obtained by either of these methods were recorded in two columns on a page, and the subtractions were performed afterwards, giving the actual width of the ring in millimeters and fractions. Thus any error in the original reading would affect two rings only. Very great numbers of readings have been done a second time and vast numbers have been checked over approximately; hence it is believed that errors of this kind are extremely rare; out of 20,000 measures, perhaps 4 or 5 have been discovered. Errors of subtraction may have occurred, but it is thought that these also are extremely rare indeed, since practically all of the work has been checked over a second time.
In the case of the sequoias, however, the method of measuring was much more highly developed. It required a cathetometer with a thread micrometer and adding machine. The cathetometer is placed horizontally on the table and the wood to be measured is also put horizontally on the table at a distance of about 33 inches. The cathetometer telescope has a lens of such a focus that 1 mm. on the wood section becomes 0.25 mm. in the focus. The micrometer has a screw-thread with a pitch of 0.25 mm., so that one revolution of the micrometer head moves the thread through exactly 1 nun. as seen on the wood. The individual measures of rings are made on the micrometer screw by reading the graduation of the head to revolutions and hundredths, giving directly millimeters and hundredths. On commencing a set of readings the stationary thread of the micrometer is first placed on the zero-year ring of each decade, and the reading of the cathetometer is made and this is entered on the adding machine. A space is then inserted on the adding machine and thereafter the micrometer reading of each ring in the decade is added in column as fast as made. Then another space is made on the adding machine and the total is entered without clearing the machine. Immediately below this total the reading of the cathetometer in the new position 10 years advanced is made and inserted on the machine without addition. Then another space on the machine is given, followed by the individual readings of the next decade. In this way all the years are read individually by the micrometer and every 10 years the sum of these readings is checked against the cathetometer reading, which should come to the same amount.
The reading of the micrometer screw to 0.01 mm. is closer than the average setting can be obtained. The rule has been generally observed that in every decade the agreement between the sum of the readings obtained and the cathetometer reading should check within 0.20 mm. In the earlier measures, where the rings were irregular or the surface of the wood uneven, this accuracy of check was not obtained in a few cases. Yet even there the error in checking was not much larger than the figure mentioned, and it is expected that the results are sufficiently close for all purposes desired. The 25,000 measures on the first group of sequoias were begun by the writer, but after 2,000 had been done they were continued by Mr. Edward H. Estill, who did them with great care. In the second group, with 22,000 rings, the measuring had been done by Mr. J. F. Freeman, who has made some slight alterations in the method above described by which an increased accuracy is obtained. As a result, the check between the decades by measure and by cathetometer is nearly always within 0.10 mm.
TABULATING.
The paper used for the tables throughout has been a cross-ruled paper with squares about three-eighths of an inch in size. This paper is 8 by 10 inches in size and suffices admirably for small tables. Usually 20 numbers are placed on a horizontal line with the beginning year at the left and with numbers from 1 to 20 at the top. Thus 1820, 1840, etc., will be placed at the left, and 1821 will be the first date given in that line. When it is desired to make longer tables, the pages are pasted together side by side or end to end, and then given a zigzag fold, so that two pages are open at once. In the case of the sequoias, with their 2,000 to 3,000 rings, no attempt has been made to paste the pages together, but enough loose sheets are used to cover the entire series at the rate of 20 years to a page. This gives sufficient vertical space to include all the necessary trees in a group and to use subgroups which may be summarized and averaged by themselves. An attempt has been made to check the addition of these numbers throughout.
AVERAGING.
In simple averaging the sums are placed in ink on the table and divided by the number of trees, using the slide rule for the process. There are several questions in connection with this subject. The first is whether straight averages of trees of widely different size give the best report of the evidence of the trees. It is evident that in taking averages of trees of mixed sizes the larger trees will carry more weight and their variations will be more pronounced in the result. But it is often the case that the smaller trees are the ones which show the greatest relative variations in the rings. This can be so much the case that the omission of a ring becomes a gross exaggeration. It is possible to use the relative values by taking the logarithm of each ring measure, averaging the logarithms, and then coming back to the number. This could be called a geometrical averaging, since it would be the equivalent of multiplying all the values together and then extracting the root equal to the number of values. In this way the small trees of the series would receive more importance. However, this plan is so long that it has not been used in practice.
One of the most common and puzzling problems is the proper method of handling the decrease in the number of trees in a group as the center is approached. A group of 5 may be selected, for example, and perhaps a century from the average center of the trees some one tree whose rings differ from the average may come to its end. It means that for 100 years near the center only 4 trees supply the data and at the point where the 5 change to 4 there is a discontinuity in the curve. In actual practice this lacking tree has usually been supplied by an extrapolation from its subsequent curve. That is, the variations assumed in the non-existent part of the tree follow precisely the variations in the remaining trees, altered to the average size of the missing tree by means of a constant factor, determined by overlapping periods. Thus, if 5 trees carried easily back to 1820, but only 4 of them extended to 1720, and it was desired to carry the full group to 1720, the period from 1820 to 1840 would be taken both for the 4 and for the 1 alone and the ratio between them determined. Now, averages for the 4 are carried back to 1720, and then the factor found in overlapping periods is applied to the mean of the 4, producing a probable value of the fifth between the years 1720 and 1820. This probable value is inserted in parenthesis in the table and all 5 values added up for an average. As a rule, groups are carried back only far enough to make assumed values of this kind a minimum in number.
There is one other problem in this immediate connection, namely, that of "gross rings." By gross rings I mean certain regions in a section where the average size of the rings becomes 2 to 5 times as great as normal. This is a problem by itself, both as to cause and as to method of treatment. Some study of its prevalence in different trees has been made, and it is usually safe to say that where an epoch is shown to have gross rings in one tree, the chances are at least even that the same years will have gross rings in the next tree. Since gross rings may not come oftener than once in several hundred years and last only 10 to 15 years, it is evident that we are dealing with something more than mere accident. The phenomenon probably has a climatic character. Yet, gross rings are not universal at any one time, and while one epoch may show gross rings in half the trees of a group it does not show it in the other half, judging by the groups examined. It is considered best to allow the ring values to enter the curves just as they are found, for while the gross rings disturb very greatly the size of a series of 10 to 20 rings, they do not seriously disturb the relation in size between a ring and its immediate neighbors. They therefore, as a rule, do not render the rings unidentifiable. It is likely, therefore, that they should be included in the means, and if some better way of handling them is discovered later it will not be difficult to apply it.
SMOOTHING.
In general the smoothing of a curve means removing some of the minor variations, so that the larger variations may be perceived. In the early part of the work the use of overlapping means was adopted. At the very start, overlapping means of a considerable number, such as 11 or 9, were used. This was quickly changed to overlapping means of 3. These overlapping means were done by the calculating machine (Brunswiga). On this machine three were added and then contin- uously the one next in sequence was added, while one at the other end of the three was dropped. However, this was changed to Harm's formula, because his formula is normally easier to apply and it gives a little more individuality to each observation. The method of applying Hann's formula consisted in adding to a table two columns consisting of, respectively, first and second intermediate values. This can be done rapidly and without taking too much space. To express the differences between overlapping means and Hann's formula graphically, we only need to say that if we take successive groups of three in any curve, forming a triangle, the center of gravity of the triangle is the value from overlapping means, but the point midway between the vertex and the middle of the base is the point from Hann's formula. In the present work Hann's formula has been used frequently, and in order to shorten description of processes the word "Hann" has been used as a verb.
In the analysis of curves already performed by the periodograph, the curves have sometimes been smoothed by Hann's formula before plotting and photographing. But a trifling error in the focus immediately smooths the curve, and therefore it is evident that the preliminary smoothing of a curve before plotting need not be done.[1]
Such preliminary smoothing helps the eye to judge variations in the curve. The effect of out-of-focus position in a photograph may be called optical smoothing. It is evident that optical smoothing may be done in two directions, vertically and horizontally. In plotting a curve it is evident that the desired smoothing must be in a horizontal direction, but in the differential photographs made with the periodograph, the directions of optical smoothing may have a very important bearing on the judgment of the significance of the photograph. Of course, in the differential pattern, long interference fringes are sought and these are emphasized by optical smoothing parallel to them. Some illustrations of this will be given under the subject of the periodograph. Perhaps no feature of this subject of tree-growth and climatic and solar variation has received more adverse comment than the matter of smoothing curves. The author is entirely open to conviction as to the advantage and disadvantage of such process, but it seems well to remember that our views as to this are likely to be a matter of convention rather than of actual thought in relation to the subject in hand. For instance, a monthly mean is a smoothed result. The rainfall, instead of being taken as it came, mostly in a few days, especially in the summer, is treated as if it were the same for every day in the month. Yearly means are smoothed values. The ordinary method of plotting yearly means is a smoothed representation of those quantities. The unsmoothed representation consists of what one may call a columnar plot. Examples of plots of that type may be found in connection with some rainfall records published by the United States Weather Bureau and in a representation of the London rainfall for more than 100 years published by the British Rainfall Association, and elsewhere. In this kind of plot the rain for a year is not represented by a dot, but by a block column which extends from the base-line up to the required amount and it has a width equal to the interval of one year according to the scale of the plot. Now, the ordinary way of representing rainfall places a dot at the middle of the top of this column, and these dots are connected together by straight lines. It is immediately seen that this cuts off each corner of the high column of any maximum year and contributes those corners to the adjacent lower column, so that the ordinary bent line of the rainfall record has thus been twice smoothed—once in the yearly sum and once in the method of plotting.
In speaking of the above records, I have in mind the smoothing in time intervals, but I would like to note also that whenever a district is averaged as a whole the average thereof is a smoothing in space. The temperature at any one time in a city station is a single definite record; but if the mean temperature in a valley or a State, for example, is tabulated, there is at once a spacial smoothing. In the minds of many students of solar variation and weather, the reason why a large group of meteorologists fail to get evidence of the relationship is because they take the average of the whole earth at once in their test of temperature changes or of rainfall. It is evident, therefore, that the reason they do not get results is because they do too much smoothing of the curves. Studies in connection with the present investigation have given some indication that small districts balance each other in their reaction to solar stimuli.
STANDARDIZING.
PLOTTING.
So many curves have been made in connection with this study that a practically uniform system throughout has been adopted. The paper used is a cross-section paper with the smallest divisions 2 mm. in extent and with heavy lines at every centimeter. The smallest divisions are uniformly used for one year unless in some special study. For the illustrations, these plots are traced and drawings made from which the engravings are reproduced. For use in the periodograph, the plot is made on the same scale and continued in length to any amount up to about 40 inches. The space between the base-line and the curve has then been cut through with a sharp knife, usually a razor blade, and the curves have been mounted in long strips some 4 inches wide and 50 inches long, and the backs painted with opaque paint. In this way they are mounted for analysis. A mirror behind reflects light of the sky overhead through the curve and supplies the necessary illumination for photography.
Problems in plotting. — In connection with the plotting of the curves used in this study, certain problems have arisen which seem worthy of consideration. The ordinary plot and the ordinary averaging seem extremely good and appropriate when the variations are small in comparison with the mean values, but when the variations are large in comparison with the mean values it does not seem to the writer certain that the usual plotting conveys an accurate idea or gives a suitable basis for further work. This may be illustrated by the plotting of rainfall. If the rainfall doubles in some unusual year, it produces an immensely greater change in the area of the curve than when it goes to one-half of the mean. Doubling the mean produces the same changes as going down to zero, though in proportion the latter is infinitely greater.
The enormous exaggeration, therefore, of excessive rain values was felt to introduce misleading material in the ordinary form of a plot. In order to overcome this at least one experiment has been made with what is called a bilateral plot. In this the quantities from 0 to 100 per cent of the mean are plotted as before, but the quantities over 100 per cent of the mean are inverted in percentage and plotted above the mean line on an inverted scale. It is recognized that this is not the perfect way of making a plot of this sort, for by this plan the mean value of the new curve will not be at the same point as before, but will be somewhat below it. However, the matter is only in the experimental stage and it has not been thought necessary to work out a correct procedure.
- ↑ The three-score of curves which are now specially prepared for examination with the periodograph carry the mean values without smoothing.