United States Department of Analyses of Great Smoky Agdcuttore

Forest Service Mountain Red Spruce Sout-n Forest Ring Data

New Orleans, Louisiana

i -%. Both mortality rate and radial growth of high elevation (>900 m) red spruce-Fraser forests of the southern Appalachians have experienced change since approximately 1960. Scientific interest In a study of these forests have increased because atmospheric pollution is a possible cause of the change. Scientists with statis tical and biological expertise independently analyzed a tree ring data set collected by the Valley Authority and the National Park Service. The objective of the analysis was to develop new or improved techniques for extracting information from such data; tree rings represent a natural data storage system that is one of the few sources of long- term information for these forests. Although no definite statements are made about the role of atmospheric deposition in observed forest decline, the results should contribute to the success of future research. The four techniques employed in the study involved: (1) a dendrochronological approach employing spline detrending and multiple regression to study the effects of climate on ring width, (2) an application of fractals to study the dependence of variance on mean ring width over time, (3) an approach that combined Box-Jenkins methods and spatial analysis, and (4) a method of studying time dependence of ring width on climate using the Kalman filter.

May 1988 Analyses of Great Smoky Mountain Red Spruce Tree Ring Data

Paul C. Van Deusen EdLzor Contents

INTRODUCTION ...... 1

Southern Appalachian Red Spr~ce--FraserFir Forests--Elizabeth Groton and Christopher Eagar ...... 2

A Tree Ring Analysis of Red Spruce in the Southern --Edward R. Conk ...... 6

Utilizing Time Series Models and Spatial Analysis of Forecast Residuals for Tree Ring Analysis of Red Spruce--2. Keith Ord and Janice A. Derr ...... 21

A Fractal Approach to Analysis of Tree Ring increments--P.A. J. Taylor ...... 40

Red Spruce Tree Ring Analysis Using a Kalman Filter--Paul C. Van Beusen ...... 57 The Institute for Quzntitative Studies at the evidence of anomalous behwior In the red spruce USOA Forest Serv.i_ce,Southern Forest Experiment forests of the and suggests Station, has been engaged in a study of that this is partly due to warmer s stiatistlhcal methods used in researching temperatures in recent years. He also staces Atmospheric Deposition Influences on Forests that the situation in southern red spruce is not AD. The siludy was funded by the ifd'ationai sfml"riar Lo that Pri rzurthern red sprcjce. Vegetation Survey, which is under the Nacio _ 1 Drs. Old and Derr performed a spatial analysis Acid Precipitation Assessment Program (NABAP). on the data. They concluded that there was a The study began with a distributed seminar; for tendency for ring wldths to diminish In recent EO weeks articles about ADIF were sent to a years and that there is a strong spatial number of participants who returned eoments each dependence in forecast residuals that cannot be week, A final report was produced (Kiester and explained by geographic or biotic factors alone, others 1985) consisting sf critiques of past ADIF They did not consider climate, which they mention studies, suggestions for additional reading, and could explain some of the remaining spatial philosophy about the type and quality of research dependence. The analysis performed here is novel needed in the ADIF area. for the field of dendrochronology and may lead to The study indicated that tree ring analyses useful results in the future. held promise for the study of ADIF, but further To study the dependence of variance on the mean development of appropriate statistical methods of the data, Dr. Taylor investigated the use of would be useful. Therefore the idea of "fractals," a term coined to denote fractional "replicated-statisticiansrrwas employed. Because dependence. He conc~uded that the change in there are several. ways to approach a tree ring fractional dimension over time may be due to analysis, it was probable that allowing a number successional, climatic, or anthropogenic of individuals to work independently would yield influences. This technique has not been applied some Lnteresting new dendrochronobogieal previously to tree ring data and nay show promise techniques. Red spruce (Pieea rubens Sarg.) was after further development. chosen for this study because claims were Dr. Van Deusen analyzed the data using the previously made that spruce forests were Kalman filter technique, which is commonly used experiencing unexplained growth declines in both in engineering applications, At the time of this the Northeast and the South (Hornbeck and Sxith study, the method had not been used in 1985; Adams and others 1985). Although the study dendrochronology, although scientists in the was intended to develop statlstieal methods and Netherlands (Visser 1986) have recently published not explanations of growth declines, using this a paper on independent applications of the method data allowed for that possibility to tree rings. The Kalman Filter allowed the This study was a joint endea- or by the USDkr Forest Service, the National Park Service (NBS), climatic data to be modeled dpamically so that and the Tennessee Valley Authority (TVA). its effect over time could be studied. He Funding came from the Forest Service and the5 TVA, concluded that these have become data from the TVA and NPS, and statistical progressively more sensitive to climate since the analyses were conducted by the Forest Service. late 1950's. This increased sensitivity may Agreements were made with the following coincide with insect-caused thinning in the scientists to perlorm analyses and provide stands. individual reports: All the studies concluded that the growth patterns in these stands have changed in the last (1) Edward R. Cook, Ph.D. 20 years. The causes of these changes are Tree-Ring Laboratory uncertain, but the sensitlivity of the stand to Lamont-Doherty Geological Observatory climate appears to be increasing. To enhance the of Colmbia University reader ' s ability to interpret the various PalLsades, New York 10964 analyses, the individual reports are prefaced by a review by EIizsrbeth Croton and Christopher (2) Keith Ord, Bh.D. and Janice Derr, Ph.D. of the geographical and biological Department of Management Science Eagar background of the Southern Appalachian Spruce Fir Pennsylvania State University Forest. University Park, PA 16802 References cited in this seetron are: Adams, (3) Robin A. J. Taylor, Ph.D H.S.; Stephenson, S.L.; BLasing, T.J.; Duvick, Department of Entomology D.N. 1985. Growth-trend declines of' spruce and 106 Patterson Building fir in mid-Appalachian subalpine forest. Pennsylvania State University En~riro~entaland Experimental Botany. 25(4): University Park, PA IS802 315-325. Hornbeck, J.W. ; Smith, R.B. 1985. Documentation of red spruce growth decline. (4) Paul C. Van Deusen, Ph.D, Canadian Journal of Forest Research, 15: 1199- Institute for Quantitative Studies 1201. Kiester, A.R.; Van Deusen, P.C.; Dell, Southern Forest Experiment Station T.R. 1985. Status of the concepts and 701 Loyola A-~enue nethodlogies used in the study of the effects of New Orleans, 'LA 70113 atmospheric deposition on forests. Internal report for the National Vegetation Survey of Dr. Cook is considered to be one of the Leaders NAPAP. Visser, ti. 1986. Analysis of tree ring in the field of dendrochronology, and his data using the Kalman Filter techniques. IAWA analysis therefore includes the most currently Bulletin n.s., Vol. 7(4) 289-297. Published at accepted techniques. He concluded that there is the Rijksherbarium, Netherlands. Soutlxern Appalachian Red Spruce--Fraser Fir Forests

Elizabeth Groton and Christopher Eagar

Host spruce-fir forests of the southern The southern Appalachian red spruce- Fraser fir+ Appalachians have been recently disturbed by the forests are found in southwestern Virginia, extensive mortality of Fraser fir caused by an western , and eastern Tennessee introduced insect, the (fig. I), Total area of these southern . This pest, a native of Europe, Appalachian spruce-fir forests is estimated at i t that feeds on the bark of true 26,577 hectares, with 19,755 hectares occurring (Abies spp.). Fraser fir is quickly killed within the Great Smoky MountaFns National Park by the balsam woolly adelgid. Mortality occurs (GSMNP), Extensive logging and other between 3 and 9 years front the time of initial disturbances early in the 20th century have depending on the size and vigor of reduced the extent of the southern Appalachian an and Speers 1965). Tree death is spruce-f ir forests. Today this forest trpe caused by the diffusion of compounds secreted by occurs on high-elevation peaks (above 1,370 m) in the adelgid into the xylem during feeding, which islandlike patches. causes formation of premature heartwood. Species ' composition in the southern Translocation of water and minerals to the crown Appalachian forests changes with the elevational are greatly reduced, causing water stress and gradient. At lower elevations (1,370-1,580m) in eventual death of the tree (Puritch 1971, 1973, undisturbed forests such as those of the Great 1977, Puritch and Johnson 1971, Puritch and Petty Smoky Mountains, red spruce ( Sarg . ) 1971). is found in combination with northern hardwood The balsam woolly adelgid was first identified type species : maple (Acer spp.) , American beech in North America in 1908 on balsam fir (Abies Ehrh. ) , yellow birch ( balsamea) in Maine (Kotinsky 1916). The adelgid tton), eastern hemlock has caused extensive mortality to balsam fir candensis (L.) Carr., northern red oak (Ouercus throughout eastern Canada; however, infestations rubra L,), Carolina silverbell (Nalesia Carolina have not progressed more than 80 kilometers L.), and yellow buckeye (Aesculus octandra inland from the coast because of extreme inland Marsh. ) . As elevation increases, the fir winter conditions (Balch 1952, Schooley and component gains importance, and forest Bryant 1978). The balsam woolly adelgid is not composition changes to predominantly red spruce- present in the northern Appalachian spruce-fir Fraser fir. Red spruce occurs less frequently at forest . the highest end of the gradient (above 1,890 m) , The balsam woolly adelgid was detected in the giving way to essentially pure Fraser fir (Abies southern Appalachians on , North fraseri (Pursh) Poir.) stands on mountain tops Carolina, in 1957 (Speers 1958). Subsequent (Whittaker 1956). Mountains that were logged surveys revealed that the adelgid had spread during,the early part of this century and did not throughout the entire 3,035 hectares of Fraser experience postlogging slash fires are dominated fir type in the Black Mountains (Nagel 1959). by Fraser fir. This includes most of the Black High mortality of Fraser fir and widespread Mountains, Balsam Mountains, Roan Mountain, and adelgid distribution indicated establishment Mount Rogers. prior to 1957, perhaps as early as 1940. Balsam Red spruce grows larger and lives longer than woolly adelgids were detected in 1962 on Roan Fraser fir, but Fraser fir grows more rapidly and Mountain (Ciesla and Buchanan 1962), and in 1963 produces more prolific seed crops than red infestations were located on spruce. Red spruce can live for over 350 years, and on Mount Sterling in the GSME;IP. grow to 40 meters in height, and have diameters The adelgid arrived in the Clingman's Dome area at breast height (d.b.h.) in excess of 1 meter, of the GSMNP in the early 1970fs, and tree Fraser fir seldom lives longer than 150 years and mortality began there in the late 1970's. attains a maximum height of 25 meters and d.b.h. Surveys found the adelgid in the Balsam Mountains of 50 centimeters. Oosting and Billings (1951) and the nearby PLott Balsams of North Carollna in found five times more Fraser fir than red spruce 1969 (Rauschenberger and Lambert 1370). The seedlings in old-growth scands in the Great Smoky balsam woolly adelgid was not found on Mount Mountains. Both species are extremely shade Rogers, Virginia, untLl 1979; however, subsequent tolerant and are capable of resuming normal stem analysis of several trees within the growth after 50 years of suppression. infested areas revealed adelgld-caused red wood beginning in 1962 (Lambert and others 1980) in the annual rings. )Redspruce-~raser fir forests will generslly By 1984 and 1985, the balsam woolly adelgid had be referred to as simply spruce-fir forests. caused extensive damage throughout the Black Mountains. Balsam Mountains, Plott Balsams,

Elizabeth Groton is forest biometrician for the Tennessee Valley Authority, Norris, TN, Christopher Eagar is ecologist for the National Park Service, Gatlinburg, TET. Figure 1.--Area covered by the southern Appalachian red spruce--Fraser fir forests. 3 Grandfather Mountain, and most of the Great Smoky spruce-fir forests of the GSmP. The second Mountains. Limited use of insecticides at Roan studyI conducted by the Tennessee Valley Mountain reduced fir rnorrality in accessible Authority (TVA) in the aut-n of 1984, areas, although nontreated areas experienced established plots throughout the range of the heavy darnage. In the Clingman's Borne area of the southern Appalachian spruce-fir type, excluding GSMNP, adelgid infestations had caused the GS-KVP, Intending to conittine data sets for significant fir mortality at elevations below future anaiyses, both agencies collaborated on i,ajC necers and minimal aamage above this sampdlng design in order to ensure that similar elevation. P40iunt Rogers had suffered the teas t data were collected. Fraser fir mortality of the southern Appalachian Problems sf assessing change in forest spruce-fir forests. There were isolated, dead produeti~rity prompted the A and NPS to Eraser fir in areas known to have been infested establish permanent vegetation plots Ln the for 23 years, but even within these areas the southern range of the spruce-fir type. Plot impact of the adelgid was surprisingly low, establishment was also influenced by the need for Possible explanations for this anomalous additional information on stand dynamics in the condition on Mount Rogers include: a genetic spruce-fir forests, This led to a collaborative based difference in defense mechanism to adergid study between the TJA, NPS, and the Forest infestation of this fir population, a reduction Service. Data collected by the TllA and NPS in the toxicity of che secretions of the Mount included tree increment core data and detailed Rogers adelgid population, or a combination of plot information, The tree core data and plot both possibilities, arized and made available to a number of individual researchers for independent analysis. Plot information included elevation, latitude and longitude, live and dead basal area, and stand density. Regional climatic data Mational Park Service and (monthly averages of precipitatsion and Tennessee Valley Authority temperature since 1933) were also included in the data. Increased mortality (Siccama and others 1382, Sampling procedures utilized by the NPS and TVA Scott and others 1984, Vogeimann and others 1985) were basically the same (table I). Both agencies and apparent reductions in radial increment used stratified random sampling, locating plots (Adams and others 1985, Bruck 1986, McLaughTin on aerial photographs and topographic maps. Data and others 1983) in the high-elevation spruce-fir that were collected from the plots included site forests of the Eastern United States prompted two characteristic data such as slope, aspect, studies in the southern Appaf achians . The topographic loeat ion, and descriptions of studies were designed to assess the current understory vegetation. Individual trees were condition of these forests, relate observe& mapped, measured, and evaluated for decline decline symptoms to site characteristics, and s3mptomology. Quantitative assessments were provide baseline data to monitor future changes made of mortality and regeneration. Increment in the forest condition. cores were collected from five dominant or The first study, conducted by the National Park ecdominant trees at each site. Two cores per Service (NPS), began in the sumer of 1984 in the tree were taken at d.b.h.

Table I.-Comparison of Tennessee Valley Authority (TIPA) and National Park Service (NPS) spruce-fir sampling procedures

Vari&le TVA NPS

Plot Location Stratified random Stratified random (Strata: Elevation (Strata: Elevation. and dry/wet) topo position, macro-aspect)

Elevation Variable within Held constant at strata a set strata

Plot Size Circular, 0.08 ha Square, 0.04 ha

Overstory stand ------data'

Site charater------istic data+

Increment cores Cores taken froni five Cores taken from five dominant or codominant dominant trees from trees from outside the within plot, cores plot and extended to not extended to tree tree center. center,

+herstory stand data and site characteristic data were basically Identical for both TVA and NPS. The results of the analyses of tree core data In: Proceedings, Air pollutants effects on may provide insight into the question of whether forest ecosystems; 1985 May 8-9; St. Paul, HH, or not the southern Appalachian red spruce and St, Paul, t4.H: The Acid Rain Foundation, Inc,: Fraser fir are experiencing a decline that cannot 137-155, be attributed to natural stresses. The plots Giesla, W.M,; Buchanan, Td,D,, 1962. Biological established by the NPS and 1111A will continue to evaluation of balsam woozly aphid, Roan be remeasured in order to monitor future changes Xountain, Gardens, Toacana District, Pdsgah in stand productivity. National Forest, North Carolina. (Forest Pest Management Report 62-93 U.S. Department of Other StudEes Agriculture, Forest Semice, Southern Area, State and Private Forestry p. 2-3. Several studies have used annual radial [unpublished]. increments from tree cores to evaluate changes in. Kotinsky, J, 1960. The European fir louse, the growth rate of red spruce and Fraser fir from Ghermes (Ratz. ) . several sites in the southern Appalachians. Entomological Proceedings Society Washington, These studies indicated an abrupt shift to narrow 18: 14-16. growth rings beginning in the late 1960's to LEunbert, W.L.; Morgan, S.W.; Johnson, K.D. 1980. early 19709 (Adams and others 1985, Bruck 1986, Detection and evaluation of the balsam woolly McLaughlin and others 1383) for red spruce and, adelgid infestations on Mount Rogers, Virginia- to a lesser extent, for Fraser fir. This -1979. Asheville, NC: U.S. Department of tendency was more drastic at high elevations Agriculture, Forest Service. Southeastern Area (Adms and others 1985). The annual growth State and Private Forestry Report No. 80-1-7. declines are very similar in tiraing to studies 23 p. done in the Northeast, but they are not as geographically widespread and are less consistent McLaughLin, S.B.; Blasing, T.J.; Mann, L.K.; within a given sample, Additionally, the Duvick, D.N. 1383. Effects of acid rain and analysis of tree ring data is extreaely gaseous pollutants on forest productivity: a complicated because of the effects of tree age, regional scale approach, Journal of the Air stand competition, climate, and physiological Pollution Control Association. 33(11): 1042- responses to stress that may persist for several 1049. years. merefore the interpretation of these Nagel, W.P. 1959. Forest insect conditions in data has been the subject of considerable the Southeast during 1958. Southeastern controversy. Station Paper No. 100. Asheville, NC: U.S. Vast differences exist in speciesf composition, Department of Agriculture, Forest Service. 10 structure, and stand productivity within the P ' limited extent of the southern spruce-fir. These Oosting, H.J.; Billings, D.W. 1951. A comparison differences are a result of climatic disparities of virgin spruce-fir forest in the northern and associated with elevationill gradients, past southern Appalachian system. Ecology. 32(1): management histories, and other site-specific 84-103. variables. These environmental factors, and the Puritch, G.S. 1971. Water permeability of the lack of historical data in the South, preclude wood of grand fir (Abies randi is (Doug.) any analysis designed to discover elther the Lindl.) in relation to infestation by the causes of observed declines or even if the balsam woolly aphid. Adel~espiceae (Ratz . ) . observed declines are abnormal in these forests. Journal of Experimental Botany. 22(73): 936- 945. Puritch, G.S. 1973. Effect of water stress on photosynthesis, respiration and transpiration of four Abies species. Canadian Journal of Adas, H. S. ; Stephenson, S . L, ; Blasing, T.J . ; Forest Research. 3(2): 293-298. Duvick, D.N. 1985. Growth-trend declines of Puritch, G.S. 1977. Distribution and phenolic spruce and fir in mid-lrppalacfiian subalpine composition of sapwood and heartwood in Abies forests. Enviromental and Experimental Botany. (Doug.) Lindl. and effects of the 25(4): 315-325. balsam woolly aphid. Canadian Journal of an, G.D. 1962. Seasonal biology of the balsam Forest Research. 7(l): 54-62. oslly aphid on Mt. Mitehell, North Carolina. Puritch, G.S.; Johnson, R.P.C. 1972. Effects of ournal of Economic Entomology. 55(L): 96-98. infestation by the balsarn woolly aphid, an, G.D. 1966. Some new infestations of (Ratz.), on the ultrastruct alsm woolly aphid in North America with d-pit membranes of grand fir, Abies possible modes of dispersal, Journal of (Day.) Lindl. Journal of Experimental conomic Entomology. 53(3): 508-511. . 22(73): 953-5358. an. G.D.; Speers, C.F. 1965. Balsam woolly Puritch, G.S. ; Petty, ffect of aphid in the Southern Appalachians. Journal of balsam woolly aphid. (Ratz.) , Forestry. 63(L): 18-20. infestation on the xylem of Abies grandis Baleh, R.E. 1952. Studfes of the balsam woolly (Doug.) Lindl. Journal of Experimental Botany. aphid. (Ratz.) and its effects 22(73): 946-952. on balsam fir, (L,) Mill. Rauschenberger, J.L.; Lambere, H.L. 1970. Status Canadian Department of Agriculture. Publication of the balsam woolly aphid in the southern No. 867. 76 p. Appalachians--1969. Southeastern Area, State Bruck, R.I. 1986. Boreal montane ecosystem and Private Forestry Report No. 70-1-44. decline in the Southern Appalachian Mountains: Ashevilfe, NG: U.S. Department of Agriculture, potential role of anthropogenic pollution. Fo-rest Service. 20 p. Sc:~oole.g, H.0.: Bryant, D.G. 1978, The balsam Speers, C.F. 1958, The balsm woolly aphid in wcio1Ly aphid in Newfoundland. Canadian Forest the Southeast, Journal of Forestry, 56(7>: Servjce Information Report N-X-160.72 p. 515 - 516. Scott, . Sicearna, Johnson, A.H.; Vogelmsnn, M,W,; Badger, 6-9,; Bliss, M,; Klefn, Breiseh, A.R. 1984. Decline of the red spruce R.M. 1985, Forest decline on CmeTs Wmp, In the Adironbacks, New Tork, Bulletin sf Vermont, Bulletin of the Torrey Botanical Tor~~yBot.a.rtLia",C1~b, 11c"L<$>. Club, L32, 275-287, awnittaker, - R,H. 1956. The vegetation of the Siccama, T.G.; Bliss, M.; Vsgelrnann, H.V. 1982. Decline of red spruce in the Green Motlntains of Great Smoky Mountains, Eeologictil Monographs, Vermont, Bulletin of the Torrey Botanical 26(1): 1-80. Club, 10912;: 162-268,

Edward R, Gook

corrected or removed from the data set. merefore the nmber of ring width series for A recent analysis of red spruce (Picea rrzbens some plots were reduced to as few as three. Sarg,) tree ring widths in southern Pi;ppalachlan After the quality check, the remaining ring stands has caused concern that the red spruce width series of each plot were standardized forests sf the southern Bippalachitm mountains may (FrFtts 1976, Gook 1985) to remove long-term be in an early stage of decline (Adams and others trends in growth associated with tree age, size, 1985)- Although many hypotheses have been and stand dynamics. The need for-standardization generated regarding the cause of the red spruce prior to creating a stand-average tree ring decline, no definite answer has yet been found chronology Is discussed in detail in Fritts (McLaugPllin h985j. LE the decline in red spruce (1996) and Cook (1987). Because the ring width ring wldth can be explained by natural effects, serLes were rarely more than LOO years long, then costly and needless pollution controls may negatlve exponential or linear regression curves be avoided. were used to detrend the series. The Tennessee Valley Authority (TVA) and the However, it was unlikely that this conservative National Park Service (NPS) conducted studies on detrending method would remove any anomalous permanent plots estiiblished throughout forests of decline signal during standardization. After the the southern Appalachians. Long-term changes in growth curve was estimated for each series, "che the composition and health of the forest were tree ring indices were computed as: monitored. Using the data provided by the NPS and TVA, the objective of my analysis was to determine if the recent patterns in the ring widths indicate an anomalous decline and if this where lit equaled the tree ring index, Rt was the decline can be explained by natural environmental ring width, and Gt equaled the growth curve factors related to climate. The analyses were value, all for year t. Therefore a tree ring performed on annual tree ring chronologies index can be defined as the ratio of the actual (Fritts 1976, Gook 1985) developed from the ring ring width to the expected value as estimated by width series of each plot. Gt. Tree ring indices have a long-term mean of 1.0 and a variance that is reasonably time stable. Thus tree ring indices are stationary processes that can be averaged into a stand- average series. After each ring width series was The quality of the data provided by the NPS and reduced to index form, the tree ring index series TVA was checked using tFe COFECW program of of each plot were averaged into a final tree ring Holrnes (1883). The program cheeked for cross- ehrsnology using the biweight robust mean dating errors, measurement errors, and other ring (Masteller and Tukey 1977) to reduce the width irregularities that mi&t Limit the influence of outliers on the computation of the usability of ring width time series for tree ring mean-value function. analysis. Because the actual increment cores were not available for this quality cheek, the program output was used to verify cross-dating, make corrections of dating when possible, and The TVA and MPS plots were stratified by eliminate ring width series for which no obvious elevation into three groups: below 5,400 feet, corrections could be made. Approximately 15 5,486 to 6,000 feet, and above 6,000 feet. These percent of the ring width series were either strata refleeted a vegetational gradient in the

Edward R. Cook is a research scientist at Lamont-Doherty Geological Obsematory irk Palisades, New Vork,

6 mountains, me lowest red spruce stratum was was much better. Only Roan Mountain and predomfnamtly a mixed -hardwoods forest ";;andfather Mountain were not represented, The zone, while the strata above 5,400 feel were rseFow-5,4QO-foot stratum was composed of 7 NPS within the spruce-fir zone (ml%e 1984), The and 22 lX7A plots, representing 93 percent of the highest elevational stratum contained the highest available plots, The geographic coverage was proportion of Fraser fir ( complete, with a11 mountains rqresented, Poir.) relative to red spruce and was the zone most heavily impacted by the balsam woolly adelgid (Eagar 1984). The elevmtLonal strata also reflected a Tree ring series invariably possess some degree climatic gradient of increasing precipitation and sf serial persistence or autocorrelation that is decreasing temperature with increasing elevation. principally due to physiological preconditioning Thus the below-5,408-footstratum was the warmest- wlthin the tree, Therefore the information and driest area and the above-6,000-footstratum contained in a given ring width is somewhat the coolest and wettest, Additionally, the determined by past tree growth and vigor, highest stratum was enveloped in clouds most Typically, the autocorrelation structure of tree often and the lowest stratum was enveloped fn ring indices can be adequately modeled as an clouds least often, autoregressive process (Cook 1985), The general me cl%m;~~$icresponse of red spruce in the autsregmessii~e (All) process of order g has the southern Appalachian Mountains has not been form (Box and Jenklns 1990) : studTed as thoroughLy as ft has In the northern P Appalachians (Conkey 1979, McT;rughEln and others %& -: et + x aizt_i 1987, Cock and others 1987), However, based on 1-1 ecological principles, it is probable that the where Zt is the observed process for year t, et role of temperature wfl% increase as a llmltlng equals an unobserved input or random shock that factor In red spruce growth at the elevatlonal does not contain any autocorrelation, and ai the extremes of the speciesi range, Precipitation autoregressive coefficients of the BR(p) process. will also be more important as a limiting factor In the context of this tree ring analysis, the at the below-5,480-foot plots, Therefore a Zt were the tree ring indices for a plot, Each gradient in the response of red spruce to climate tree rlng series was modeled as an M(2) for the should be found that will correlate well with common internal 1930-83, The chobce of an AR(2) some aspect sf the know climatic gradient, model was based on previous experience modeling The stratiflcatlsn by sievation produced 11 TVA longer red spruce chrono%agies as Mt processes, and 8 NPS plots in the above-6,000-footstratum, The common St persistence structure among 811 $3 TVA and 6 NPS plots in the 5,480- to 6,060- plots within each stratum was also estimated foot strat-m, and 24 WA and 7 NPS plots in the using a pooling procedure described in Cook below-5,400-footstratum, At this stage, some of (E985), DTfEerences between the eom-ron Ba_% model the plots were eliminated because of the and those for the individual series were useful shortness of the tree ring chronologies. Because tools for measuring the level of autocorrelated 1930 was chosen as an initial criterion for the noise in the 1ndivLdual serPes, which may have inclusion of tree ring series in the analyses, been caused by different stand histories and any series beginning after 1930 was eliminated. disturbance regimes, The year 1930 allowed for the Bnclusi-m of the For the above-6,000-footstratum, the eomon or large majority of plots and siarultameously pooled AE esefflcients and the ercent variance provided an adequate time base for the explained by autoregression (R8 ) were: ail = dendroelimatic analyses. A longer time base 0.461, @2 - 0.284, and - 46.1 percent. For would have been better, but it also would have the 13 individual series, the average statistics eliminated too many sites. firtherrnore, the best were: Qi.1- 0.566, 32 - 0.138, and Et2 - 47.9 available cblmatic: data began in 1931. percent. Although the Et2's were sFailar, the After the elimination of short series, the coefficients were noticeably different, probably sanrple depth of the 1930 decade was examined for because of residual trend or trendline lack-of- each remaining chronology. If the sample depth fit in the individual series. However, the was Largely based on only one increment core in similarity of the ~~'ssuggested that the that decade, then that chronology was differences between the tree ring chronologies eliminated, The reason for this step in the were largely random through time, Thus the bong- preliminary screening was to Lnsure chat the 1930 term disturbance histories sf these plots may decade would not be unduly affected by poor have been similar since 1430, replication at some sites. For the 5,400- to 6,000-foot stratum, the The final result of the screening was the pooled statistics were: = 0.324, = 0.177, selection of 13 plots above 6,080 feet, 15 plscs and Ft2 = b8,1 percent. For the 15 rndividuai between 5,460 and 6,000 feet, and 29 pLsrs bel3w series, the averag? statistics are: @I = 0,484, 5,400 feet, Eight above-6,000-footseries were = 0.168, and EZL - 39-1 percent, The pooled from NPS plots, four were from TVA Mt. !+Iitchelk, AR(1) coefficient and lt2 were considerably and one was from TVA Roan Mountain. Thus the smaller than the average values for the above-6,000-foot series comprised 68 percent sf Individual series. The Latter indicated a high the available plots. The geographic coverage of level of autoeorrehated noise or out-of-phase this stratum was obviously limited by the maximlun behavior between series. Theref ore the elevations of the nountahns, The 5,400- to disturbance histories of these plots were 6,000-footstratum was earnposed of six NPS plats probably more variable than those in the higher and nine TVA plots; these represented 78 percent stratum, of the available plots. The geographic coverage For the below-5,490-foot stratw~, the pooled statistics were: = 0.422, @Z - 0.124, and R~ 13 plots, the first eigenvector of the original - 24.4 percent, For the 29 individual series, tree ring indices accounted for 50.7 percent of the aqrerage statistics were: 451 - 0.530, @a - the total variance, while that of the prewhhtened 0.044, and R2 - 39.3 percent. These statistics indices accounted for 53.6 percent of the were close to those from the intemediate variance. Cornon variance increased after elevation plots and indicated a similar degree of prewhdtening because of a reduction of noise nonhonrsgeneity from plot to plot. variance resultant from autoregressive modeling, The Lower levels of plot homogeneity in the In figure 1, the loadings for this eigenvector strata below 6,000 feet suggested that these are all positive, indicating an existLng comon plots have more varied disturbance histories, An signal iunong all series. Tke loadings for the examination of the individual the? series from prewhitened Pndiees were more unifomly positive these plots confirmed this inference. Some of than those of the original indices. This the plots showed release patterns early in this uniformity indicated that some of the differences century that were consistent with logging between the original indices were mplified by activity. Given the much reduced spatial the autoregression within those series, coverage of the dove-6,000-foot plots, the For the 5,400- to 6,000-foot stratuni of 15 higher Level of homogeneity among these plots was plots, the first eigenvector of the original tree probably related to the lack of interference by ring lndiees accounted far 34,l percent 06 the man. variance, while the prewhitened indices accorrnted for 45.3 percent. The larger increase in cornmon S SIS (PCA) variance after prewhitening indicated that these chronologies were less homogeneous than those in Because each chronology was based on a small the higher stratum. Greater variability in site sample of trees, the dendroclimatic modeling of characteristics and stand histories in this :h plot chronology was not considered a viable intermediate stratua may have caused the approach. The results would have been somewhat difference. Comparing the single series and chaotic because of the very high level of noise pooled autoregression models also indicated the in each chronology. Therefore the eomon lack of homogeneity between the chronologies. variance among all series within each stratum was However, the more restricted geographic coverage pooled using principal components analysis (PGA) of the high stratm may be a biasing agent in (Cooley and Lohnes 1371). In PCA, the structure this comparison, As before, the eigenvector in the correlation matrix of variables is loadings (fig. 1) were also more uniform after trmsfsrmed into a new set of uncorrelated or prewhitening. orthogonal modes of behavior called For the below-5,400-foot stratm of 22 plots, eigenvectors. Each eigenvector accounts for a the first eigenvector of the original tree ring unique proportion of the cotal variance in the indices accounted for 32.6 percent of the original data. The first eigenvector associated variance, while that of the prewhitened indices with the largest eigenvalue accounts for the accounted for 40.7 percent. The magnitude of the greatest percentage of comon variance among all difference was similar to that of the variables in the correlation matrix. intermediate stratum. Therefore the level of Each eigenvector is composed of a number of homogeneity between plots was similar, an loadings or coefficients equal to the number of inference also supported by the earlier original variables in the correlation matrix. autoregressive modeling results. The eigenvector These loadings, which may be positive or loadings (fig. I) of the prewhitened indices were negative, reflect the relationships between also more uniform than the loadings of the variables for a specific eigenvector. Frequently original indices. the loadings of the first eigenvector are all Generally the strength of the comon signel positive or negative, which indicates that the within each stratum was directly correlated with variables being analyzed all behave similarly. the elevational gradient. The tree ring patterns Therefore, in this study, the tree ring series at of high plots were more similar mong themselves each elevational stratum could exhibit a comon than those of the lower plots. Although it may signal due to climate, disturbance, or pollution. appear that this result reflected more limiting The loadings of the first eigenvector can also growth conditions towards the upper elevational be used to create a time series of scores that limit, the bias in the geographic coverage of reveal how this most colaraon component among all that stratunn limits the strength of this series behaves through time. This series of interpretation, scores is similar to a weighted mean, because The eigenvector mplitudes or scores of each each series is weighted by its eigenveetor stratura are shown in figure 2, The solid line Loading and then s ed with the other weighted plots were derived from the orignal tree ring series for each year. The weighting scheme is indices, and the dashed line plots were derived optimal in the sense that no other eonrponent can from the AR(2) prewhitened indices. account for more of the comon variance between The scores derived from the original indices series than the first eigenvector. Consequently, indicate an overall pattern of below-average the scores for each elevational stratum should growth at all plots since about 1966. The have a strong conunon signal for dendroclimatic largest departure was for the above-6,000-foot analysis. plots. The average score since 1966 was -2.65 The PCA analyses were done twice for each with a standard error of 20.495. For the 5,400- stratum: once on the original tree ring indices to 6,000-foot stratum, the average score was- and again on the indices after removing AR(2) 1.54 f0.522. And, for the below-5,400-foot persistence from each series. Indices are stratm, the average score was -1.73 20.638. referred to as prewhitened after removal of AR(2) These long-term departures appeared to exceed the persistence. For the above-6,000-footstratw of 95-percent significance level using a simple t- test. However, the use of a t-test on WPB mD. TVA BED SPRUCE PM)T

0.0

NPS MDWA RED SPRUCE: PLXZT

0.0

WP9 aii) T1IA RED SPRUCE PWT

Figure 1--The eigrnvector loadings of the firsr principal component of each red spruce elevarion srntum. The solid line plots correspoid ro che loadings of ihe original tree ring index chronoiogies. The dashed line plots correspond to ttie Loadings of the same series after second-order autoregression has been removed from each one. The eigenvectors were exrracted from the correlation matrix The percent variance accounted for by each eigenvecror, original jprewhi tened) , is indicated. Figure 2.--Theprincipal componenC amplitudes or scores corresponding to the ekgenvectors in figtire 1. The sdid line plots are fir the original series. The dashed line plots are for the prewhitened series. autocorrelated time series such as these can be intempention around 1966 changed the system, the extremely misleading. The number of Endependent timing of the Enternention was hypothesized only observations and the degrees-of-freedom may be after an examination of the data. Exus any mch less than the number lndieated by %Re inferences concerning a change in persistence avaf iable obsematlons . structure because of an inte~~entisnmust cake One way of avoiding the negative effect sf into account the a posterLori,naturc of these aaa-,o~crrsfatiozOF, the Ogc;greeq-oi-freedom 19 to analyses This issue will, be addressed later, as use the scores of the -AR(2) prehitened indices, it affects significance tests, By definition, these scores do not have any The results of the intervention analysis were significant autocorrelation related to that level as E0l10'lr;s: of autoregression, In figure 2, the grewhitened scores indicate a smaller reduction in growth since 1966 in all series, The post-1965 means confirmed this, For the above-6,000-foot,5,408- Above 6,000 ft 6.268** -0,566*** 58,138 to 6,000-foot,and below-5,400-foot strata, the 5480-6000ft 0,234* -0,369** 27-68 1966 to I983 means were -1-09 k0.779, -0.42 k0.807, and -0-71k0.992, respectively, None sf these means passed a t-test at the 95-percent sdgnificance level, Therefore there may have not "seen any reductfon in growth since 11966, The The strength of the step intervention was decline in the original tree ring indices may be directly correlated with eleva~~Lrsn. The above- largely explained by the endogenous 6,000-footscores showed the strongest indication autoregresshe persistence of the tree ring data of an internention in 2966, which resulted In a and the way in which it amplifies the behavior of steplike reductLon in growth. Thls result was the random shocks, et, which are largely esnsisrent with the original examination of the exogenous to the plots, 1966 to I983 means for these scares. However, The above conclusion was conservative because the step-elevation relationship was new. The the AR coefffcfents used for prewhltenlng were h2gh level of persLstence In the above-6,000-foot based an. information in both the pre-1966 and scores was greatly reduced by the step. In post-I966 time periods. The method einployed contrast, the persistence in the bower strata was minimized the probability sf a type I error reduced less. The reduction in persistence from because It assSmed that the coefficients had the earlier m(2) modeling appears to be not changed through time despite an intervention proportional to the strength of the step. in the et that may have occurred in 1966, Since The probability Levels of the intervention an Internention in the et could have a strong analysis were based on an a prior% significance impact on the estimation sf the AX coefficients, test In eaeh ease. Acceptance of these results the prewhitening may have remo*red part of the would effectively minimize type IT error at the response to an hntementlon bad it occurred, In expense of type I errsr, In contrast to the order to reduce the probability sf a type I1 earlier prewhitening results that minimized type errsr in these analyses, an alternate method sf 1 error. Therefore these sets of results served intervention analysis (Box and Tiao 1975) testlng as useful limits. As noted earlier, there was a for the occurrence of an intervention in problem In applying a prior% significance tests autocorrelated time series was used, to a statistical analysis problem that was principally based on an a posteriori examination of the data. Furthermore, the a posteriori examination of the scores for an intervention Internention analysis specifically allows for allowed for 50 possible intervention dates for a autoeorrefati_onwhen testing for the oceurence of step-function. an intervention in time series. A simple form of Based on probability theory, the probability of intervention analysis was used in this study to finding a statistically significant step- test for the occurrence sf a decline in the fuction under such conditions is related to the scores since 1966. The form of the intervention a prior2 significance level as: chosen was a step-function, which is expressed as [O 0 0 . . . 'j from 1930 to 1965 and [I E Z . . . ] from 1966 to 1983. This step-function served as one of the predictor variables in the analysis. where P is the a posteriori probability Level, p To account for autocorrelation in eaeh time is the a priori probability level of the test series, the scores for years t-l and t-2 were being applied, and rn is the nmber sf times the also used as predictors, The model used allowed tesr: esukd be applied to the data. If chis for both the occurrence a& ti step reduction in correction is applied to the probability levels growth persistence, and AR(2) The grrtervention for the step interventions shorn above, only the model was set up as a multiple regression above-6000-foot step intervention remains analysis problem. In each case, onby the step- statistically significant (P

Figure 3.--The actual (sofid) and estimated (dash) tree ring scores based on fitting a step intervention model to each series. The date of the intervention is 1966. elevation, The eause of the growth reduction is time. The degree to which the gradient exists still undetermined. Climate or other changes in probably depends on which climatic variables are natural influences are as probable a eause as are limiting to red spruce growth in a given year and anthropogenie pollutants, how those climatic variables are influenced by The realfty of the elevational gradlent in step elevation. For exilmple, based on the physics of size may be questloned. However, the existence precipitation fomatlon and its interaction wieh of L-nowr~erivirsrmental-climatic gradients in the orography, the influence of droqght on red spruce muntains suggests that the step-size gradient is growth should diminish with increasing elevation. a reality, The step-size gradient may be due to Wowexrer, once the available moisture supply is no a temperature-related phenomenon. This longer limiting to growth, this drought response hg.~"athesisis consistent with what is now known gradlent would probably disappear from the tree about the relationship between temperature stress rings. and xed spruce declines in the northern In this study , dendroclimatie modeling is Appalachian Mountains (Cook and others 1987). limited by the lengths of the series being modeled and the unavaklability of long ellmatie time series, Ideally, the modeling should proceed as described by Cook 1987; the dendroelimatic signal shasuld be modeled for a long preintervention time period of perhaps 50 to As noted earlier, the three strata used in this 60 years. A model should then be used to study follow both vegetational and climatic forecast or predict tree rings through a period gradients, which are directly correlated with to the present that includes both another elevation, To illustrate the refkectlon of this preintervention time block and the post- gradlent in the tree rfngs, figure 4 shows the interventlolil perlod. The time stability of the three series of original tree ring index scores dendroclimatie model must be tested; therefore sblperiaposed on eaeh other. Various time periods another grelntervention time period is needed. in figure 4 (1932-34, 1935-57, 1959-63) indicate if the model is verFEied as time stable, then it striking gradients across scores correlated with can be used to test for an intervention that elevation, The presence of these gradients changes the tree ring response to the model. across scores suggests that an elevationax This method has been successively used in gradient in the climatic response of red spruce analyzing the red spruce decline in the operates at times. &palachian Mountains (NcLaughlin and others At other times In the scores (1940-42, 1958, 1987, Cook and others 1987). 1969), the gradiect breaks down, and the scores Beearzse of the insufficient tree ring time of all strata are sirnLLiar, This similarity base, a weaker method of modeling was Implemented suggests that the relationship between cllmriltic that provided a basis for inference regarding the response and elevation is noastationary through climatic response gradient hypothesized earlier,

Figure &.--The amplitudes of the original tree ring indices superimposed on eaeh other. The purpose of rhis g2oi: is to highlight certain time periods when elevation-related gradients in climate response are likely to be occurring. This method was based on simple correlations latter problem may also indicate a loss of between the tree ring scores and monthly climatic climatic signal comparable to what has apparently data for the period 1931-83. happened to the declining red spruce in the Prior to the correlation analyses, the tree northern Appalachian Mountains (Cook 1987, ring scores for each ele-rational stratum were McLaugklin and others 1987, Cook and others, prewhitened to remove autocorrelation. In each 1987). case, signif5tarzt k&?(1) or AR(2) persistence was "Ln the sulte of 96 tota"lcsrreiatiorts, only removed. The monthly ternperature and three monthly tentperature variables showed any precipitation data, averaged over the northern consistency through both time periods; July, and southern mountain climatic divisions of North August, and September temperatures correlated Carolina and the southwestern mountain division with t+l lagged tree rings as follows: of Virginia were similarly modeled for autocorrelation. In this case, the climatic data showed very weak or nonexistent autocorrefation Stratum 1932-62 1966 - 80 out to lag 3. Therefore, the climatic data were not prewhitened. Above 6,000 feet July -0.343* The dendroclimatic modeling was then treated as August -0.268 z multiple input - single output transfer function Sept . -0.223 (Box and Jenkins 1970) in which ring width was a 5,400-6,000feet July -0.388** function of climate. Given the lack of August - 0.334* autocorrelation in either the input or output Sept . - 0.403** series, the principal aim of the transfer Below 5,400 feet July -0.351* function model was to identify those climatic variables that correlated significantly with tree rings and identify any delay or lag-response between the inputs and the output. The analysis assumed that the climatic variables were orthognal, an assumption that was violated for almost a11 ~~ariables. This violation may have There is an indication, especially in the 1966-80 increased the number of significant climatic period, of an elevational gradient in the variables in the model. However, since the aim response to the temperature variables, The high- of these analyses is strictly correlative and not elevation stands seem to be less sensitive to predictive, this should not have any serious summer temperatures than the lower stands. There impact on the results. is also an indication that the below-5,400-foot The tree ring scores were lagged up to 3 years spruce have been more sensitive to smertime in the transfer function analyses, meaning that temperature since 1966. each monthly climatic variable was correlated On the basis of these monthly temperature with each series of scores for years t, t+l, t+2, correlations, the July, August, and September and t+3. A plot of the correlations by lag temperatures were averaged into a summer season produced a normalized form of the impulse temperature series (fig. 5). Of particular response function for 1932-62 and 1966-80 time interest is the smer of 1980, the warmest periods of the system being modeled (Box and summer in the southern Appalachians since 1931. Jenkins 1970). For each period, 48 precipitation According to figure 4, the poorest growth year and 48 temperature correlation coefficients were for red spruce at all elevations was 1981. Given computed. While the a posteriori nature of these the t+l lag response of red spruce to summer analyses makes the use of a priori significance temperatures indicated above, the poor 1981 tests very questionable, these results should be growth year was probably related to excessively viewed as more exploratory than confirmatory. warm summer temperatures in 1980, which extended Therefore the a priori confidence limits will be to the highest elevations in the mountains. used to assess the significance of the Linear regression analyses of the smer correlation coefficients. tenrperature series versus prewhitened red spruce The results of this modeling were somewhat scores indicated that the below-5,400- and 5,400 complex to explain. In each time period, some to 6,000-foot strata were equally sensitive to indications of climatic gradients were found, previous summer temperatures over the period For example, in the 1932-62 period, the 1932 - 83. The regression R~ ' s were, respectively, correlation between tree rings and March 0.199 and 0.182. In contrast, the above-6,000- precipitation at lag t+2 were: foot regression It2 wpls 0.137. The prewhitened scores and their temperature estimates are shown Above 6,000 feet: -0.102 in figure 6. Interestingly, all strata follow 5400 to 6,000 feet: -0.604 the pattern of s er temperature almost Below 5,400 feet: -0.636 perfectly since 1978. This corresponds to the warmer than average temperatures since 1977. The correlations of the lower two strata were It was previously suggested that the lower significant (p<0.001) in a statistical sense. elevation plots should be more stressed by However, the t+2 lags were very difficult to precipitation deficiency than the higher explain physiologically. More disconcerting, elevation plots. In order to determine the these correlations completely lost statistical degree of drought sensitivity in the prewhitened significance (maximum lr1<0.15) in the 1966-80 red spruce scores, the monthly Palmer Drought period. Therefore these significant correlations Severity Indices (PDSI) (Palmer 1965) were were either spurious or the climatic signal in computed from the divisional average temperature the red spruce was highly nonstationary. The and precipitation data. Simple correlations were JULY-AUGUST-SEPTEMBER AWUGE TEMPIFMTUIIIES

Figure 5.--July,August, and September average temperatures for the southern Appalachian Mountains since 1931. The dashed line highlights variance at frequencies of 1/20 year or less. again computed between monthly PDSI and tree ring the time period of 1952-55, which contained the scores at lags t through t + 3 for the time worst drought in the southern Appalachians since periods 1932-62 and 1966-80. The results of 1931. The very strong elevation-related gradient these simple correlation analyses were very in the tree ring scores for this period was similar to the analyses reported earlier; the almost definitely caused by severity of this correlations were generally not time stable. drought and the way in which it diminished with However, as before, summertime drought and t + 1 increasing elevation. It is difficult to explain lagged scores showed some time stability and why red spruce at all elevations showed statistical significance. approximately the same level of response to PDSI since 1966. Given the shortness of this time These correlations were as follows: period, it is possible that these results are ques tionabl-e, even with the high significance Stratum Month 1932-62 1966 80 levels of the correlation coefficients. However, - this apparent increase in sensitivity to Above 6,000 feet July summertGe moisture availability should be August investigated more fully, as better statistical Sept . methods and tree ring data become available. 5,400-6,000feet July Linear regression analyses of the prewhitened Augus t scores versus summer PDSI for the period 1932-83 Sept. indicated a weaker relationship overall than for Below 5,400 feet July summer temperature alone. The R~'S for the August below-5,400-,5,400- to 6,000-,and above-6,000- Se~t. foot strata were 0.151, 0.114, and 0.035, respectively. The actual and predicted scores from these models are shown in figure 8. There generally appears to be less time stability in In the 1932-62 period, there was a clear PDSI-spruce relationships. indication of an elevational gradient. The er temperatures and PDSI were correlated highest stratum showed no s (re-0.38) because the temperature data were signal in contrast to the lower strata. However, partially used to estimate the PDSI's. However, all strata showed a very pronounced swertime the level of correlation was not high enough to drought response in the 1966-80 period. This indicate that the PDSI correlations were apparent increase in sensitivity to PDSI was completely confomded by the temperature effects. stronger than that indicated by sunnmertime In fact, when the PDSI and temperature variables tentperature alone. were used in a stepwise multiple regression As before, the monthly PDSI' s were averaged analysis to predict red spruce scores, each into a summertime season estimate of drought variable entered the model according to the since 1933. (fig. 7). Of particular interest is strength and sign of its original correlation Figure 6.--Actual (solid) and predicted (dash) prewhitened tree ring scores. The sumer temperature series (fig. 5) was used as the predictor of tree rings. The of each model is indieaced by the RSQ value. JULY---AUGUBT-SEPTEMBER ER DROUGHT INDEX

Figure 7.- -Jufp , August , and Seg tember average Palmer Duough t Severi ry Indices (PDSb) series Par the southern Appalachian Plountains. The dashed line highli;ghrs variance at frequencies less than 1/20 year, with the scores. The resulting sfor the others 1987, Cook and others 1987). However, bebow-5,400-, 5,400- to 6,000-,and above-6,000- based on the temperature modeling demonstrated in foot strata were 0,252, 0.203, and 0,123, this study (fig. 6), the dendroelimatlc signal respectively, The actual and predicted scores appeared to continue through the post-I965 from these models are shown in figure 9. decline period, Thus the reported decline does not seem to represent a major loss of tree vitality as was indicated for the northern red spruce. At this stage of inquiry, the The results of this study indicated tha% red Appalachian Mountain northern and southern red spruce in the southern Appalachian Momtains have spruce situations appear to be different. exhibited, to varying degrees, some irregular The dsndroclimatic analyses revealed that behavior in their ring widths since the mid- prevf ous summer temperatures correlated 1960's. At elevations above 6,000 feet, significantly with red spruce ring width the statistical evidence suggest a steplike decline following year. The lag-l negative temperature in radial increment since about 1966. This correlations were remarkably consistent with more decline has not been correlated with any specific rigorously delreloped dendroclimatic models for climatic deviaeion In this study. However, the srmerous stands sf red spruce in the northern way in which the magnitude of the decline Appalachians (McGlaugklin and others, [in press], increased with elevation suggested that the cause Cook and others [in press:), It is increasingly of the decline was somewhat rebated to elevation, clear that the sole of previous s A more thorough search for natural and temperature as a determinant of red spruce growth anthropogenie causes of this putative decline is and vigor is g warranted. In additfon, new and improved iqortantky, warm s collections of ring width data are highly be strongly correlated with past and present desirable to refine the statistical analyses and decltnes of red spruce in the northern validate or refute the intervention results Appalachians (Cook and others 1987). Shoul presented here. apparent increase in sensitivity to prior-s The decline of the southern red spruce at high temperatures be correct for the below-5,408 elevations could lead to broad scale mortality, spruce, low-elevation spruce in the southern as found in northern Appalachian stands. are likely to decline if warmer than However, the dendroclin~aticmodeling has revealed er temperatures persist. A warmer an apparent singular difference between the by C02 and other greenhouse gases northern red spruce and southern red spruce would not positively affect the future of red conditions in the Appalachian Mountains since the spruce in North America. 1960". In the northern red spruce, the The apparent increase in red spruce dendroeiimatic signal completely disappeared sensitivity at a11 elevations to drought since after the trees entered the post-1960 period of 1966 and with sensitivity to s declining ring widths (Cook 1987, McLaughlin and since 1977, suggests that the southern red spruce ACTUU, AND PREDICTED W0-em-FOOT SCORES -- BSQ =&llC

Figure 8.--Actual (solid) and predicted (dash) prewhitened tree ring scores. The sumer PDSI series (fig. 7) was used as the predictor of tree rings. The R' of each model is indicated by the RSQ value. PIEtPtTPImI) Urn- BE;"OmB-- Ii9Q =&PZ3

-& ldss ldra lei5 lab 1& 1k 1 18;o 1 ldso TEAR

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Figure 9.--Actual (solid) and predicted (dash) prewhitened tree ring scores. The predictors in the multiple regression analysis were sumer temperature and sumer drought. The of each model is indicated by the RSQ value. *are in a prolonged period of climatic stress. A agricultural ecosystem; 1985, May 12-17. similar pattern of increased climatic stress from Toronto, Canada, Springer-Verlag. 277-290. about 1938-60 preceeded the current broad scale Cook, E.R,; Johnson, A.W.; Blasing, T,J, 1987. decline of red spruce in the northern Forest decline: modeling the effect of Appalachians ([Cook and others 1987). Presently climate, Tree Physiology. 3: 27-&0, it is Impossible to say that this circhurastantiaf Eagar, 6. 1984. Review of the biolo@;gr and agreenent in swptomology is part of the ecology of the balsam vaskZy aphid tn the epidemiology af red spruce decline, However, it southern Appalachian spruce-fir forest. In: is cause for concern and warrants further study. mite- P.S., ed. The southern Appalachian spruce-fir ecosystem: its biology and threats. Resource/Wesearckr Managemenrr Report. SEE-71. U,S . Department of Interior. National Park Service - Southeast Region. Fritts, W.C, 1916, Tree rings and climate, New Adams, N.S.; Stevenson, S.E.; Blasing, T.J.; York: Academlc Press. 567 p. Duvick, I). N. f 985. Growth-trend declines of Wolmes, R.E. 1983, Computer assisted quality spruce and fir in the mid-Appalachian Stabalpine control In tree-ring dating and measurement. forest, Envlroment and Experimental Botany, Tree-Ring Bulletin. 43: 63-78. 2513): 315-325. Johnson, A'H.; Siccama, T.G. 1983. Acid Box, G.E.P. ; Jenkins, G. 1970, Time series deposition and forest decline. Environmental analysis: forecasting and control, San Science and Technology. l7(7): 2948-305A. Francisco: Holden-Dq, 553 p. McLaughlin, S.B. 1985, Effects sf air pollution Box, G.E,P,; Tiao, G.C. 1975. Intervention on forests: A critical review. Journal of the analysis with application to economic and Air Pollution Control Association. 35(75): 511. envlrormenttsl problems. Journal of Arneriean McLaughlin, S.B.; Doming, D.J.; Blasing, T.J,; Statistics Association. 70(149): 70. Cook, E.R.; Adarns, H.S. 1987, An analysis of Conkey, L.E. 1979. Dendroclimatology in the climate and competition as contributors to the Northeastern United States. Tuscon, AZ: decline of red spruce in the high elevation University of Arizona, 78 p. P4.S. thesis. Appalachian forests of Eastern North America. Cooley, W.W.; Lohnes. P.R, 1971. Multivar-iate Oecofogia. 72: 487-501. data analysis. New York: Wiley i4. Sons. 364 p. Palmer, W,G, 1965. Meteoroiogical drought, Cook, E.R. 1985. A time series analysis approach Weather Bureau Research Paper Nwbex 45, U.S. to tree-ring standardization, Tucson, AZ: Department of Csmerce, Washington, DC. 58 p. University of' Arizona. 171 p. Ph.D. White, P.S. 1984. The southern Appalachian dissertation. spruce-fir ecosystem, and introduction. Cook, E.R. 1987. The use and limitations of White, P.S., ed., The southern Appalachian dendrochronology in studying the effects of air spruce-fir ecosystem: Its biology and threats. pollution on forests. In: Proceedings of the Resource/Researeh Management Report SER-71. NATO advanced research workshop: Effects of U.S. Department of Interior, National Park acid deposition on forest, wetlands, and Service - Southeast Region. UtZlizfng Time Serdea Models an 5s of Forecrrr;lit Eeafhls Eon Tree jX%w Spmee

3, Keith Ord and Janice A. Derr

The informaclon from a field study on permanent Step 1.--Construct an average ring width time plots established by the Tennessee Valley series for each of the study plots established by Authortty in the Greae Smoky Mountains was used the WA, to detect and evaluate recent changes in annual ring width of red spruce (Picea rubens Sarg,). The following are decisions made during the Time series models were fit to mean annuah rLng exploratory stage of the analysis: widths of a maxLmwn of 5 mature red spruce trees Choice sf Plots.--Theanalysis reported In this for each of 44 plots for the years 1900-84, =e proj ee t concerns 48 experimental plots mean level a6 resicZuals from forecasts for the established by the mA in the Great Smoky Past 20 years sf the series were generally Mountains in North Carolina. The decision not to negative, indicating a reduced ring width include the experlnental plots established by the relative to predicted ring width. These forecast NPS in the same general region was motivated by residuals showed substantial spatial dependence time and resource constraints. that could not be explained by geographical factors alone, blhen both geographical and biotic Choice sf Measurement Scale.- -Graphs of the factors, primarily measures of stand quality, time series for each of two cores taken from five were taken into account, the residual variation trees usually produced very similar patterns. in ring widths showed a weaker pattern of local SFnce the overall plot was the focus for this sphtial dependence. study, the two core serles were averaged for each tree, The selection of a designated time period as a series for the entlre plot was more dlffieult, since individual trees may shew considerable variations from year to year. The The Katioxal Park Service (NPS) and the overall mean was chosen as the measure sf average lenriessee Valley Authority CTVA) oandueted ring width for the plot, Further eonsideration studies on experimental plots in the Great Smoky sf this issue is presented in the Discussion Mountains, produci~ga substantial data base of section. information that can "c used to examine annual ring widths sf red spruce (Pica rubens Sarg.) . Selection of Trees and Time Frame for In this report we discuss one approach to Analysis.--Graphs for the time series of ring analyzing ring widths, The objective of our widths for each of the 5 trees per plot were then study was to deteet recent changes during a constructed for all 48 plots. Examples of four designated time series (1900-84) that may be of these graphs are shorn in figures I through 4. attributable to enviramental changes, such as From an examination of the 48 graphs, the the occurrence of acid deposition. Variations in following decisions were made: ring widths relative to historical patterns are assessed. Also described is how to determine 1. Only red spruce would be used in the data whether such patterns exist because sf plot analysis to remove some heterogeneity from characteristics or additional spatial effects. the time series of ring widths averaged The main steps of the study may be s across trees in a plot. Saqle size was as follows: not seriously reduced because 214 of 234. trees in the study were red spruce, and 1, Construct an average ring width tLme series only 4 plots had no red spruce. far each of the study plots established by Only red spruce trees with a pith date the ??$A. Plots established by the NPS were earlier than 1940 would be included in the not included. study. Therefore 25 red spruce trees were Develop measures of recent increases or eliminated, and some heterogeneity caused decreases in ring width for each plot by an apparent initial rapid increase in relative to values, forecast ring width in the early years sf growth was increases or decreases in ring Re-Late the alleviated, width to geographical and biotic factors, 4. Determine whether there is any spatial 3. Wing widths from 1900 onward were analyzed, pattern to the values of excess or The heterogeneity caused by the staggered deficiency and whether geographical and entrance of trees into the plot averages biotic factors are responsible for the and by the apparent initial rapid increase pattern. in ring width was minimized,

J. Keith Ord is a professor in the Departments of Management Science and Statistics. Janice A. Derr is managing director of the Statistical Consulting Center, Pennsylvania State 'ilni-~ersbty , University Park. Pennsylvania. TVA PLOTS ;P 'rJVE'-IZ VE3AlvS CF -'#Q 5,3Es ":Gi; TREE p~r"i;=s

Figure 1.--Graph of ring widths of red spruce {Picea rubtzns Sarg.) on plot 6 of 48 selected experimental plots established by the Tennessee Valley Authority in the Great Smoky Mountains. TVA PLOTS I".RITH~&E~ICbA.PIEII;bS GF TWO SIS;tS EPGh TREE @tGT.=;I@

Figure 2.--Graphof ring widths of red spruce (Picea rtrbens Sarg.) on plot 68 of 48 selected experimental pLots established by the Tennessee Valley Authority in dhe Great Smoky Mountains. TVA PLOTS ARITHMETIC MEANS CF T'&G SICES EAZY "PEE PLc"-23

LEGEND: TREE 1 2 3 -4

Figure 3.--Graph of ring widrhs of red spruce (Picea rubens Sarg.) on plot 23 of 48 selected experimental plots established by the Tennessee Va1Ley Authority in the Great Smoky Hountains. TVA PLOTS ARITYkdET~C MEANS OF TWlfi SIDES EACH TREE PL13T-3'1

Figure 4.--Graph of ring width of red spruce (Picea rubens Sarg.) on pLor 31 of 48 selected experimental plots established by the Tennessee VaiLey Authority in the Great Smoky' Mountains. Choice of Explanatory Variables,- -The plot (48 minus the 4 wzth no red spruce) from the TV7A characteristics that were used in the study study. The time serles began with the year 1900 lnclcded both geographical and biotic factors as and ended with 1984. The series included on-by measured by survey teams, Average annual chose red spruce trees with pith dates earliar temperature and total annual precipitation for than 1948, Each time series entry was an average the three climatic regions in the study (Worth of the width of two cores from each eree, taken Carnlizsa nort%ern morintaS nc: North Carol ina cron 3 vaxfm*~~3f five red sprxce trees. Table 1 southern mountains, and Virginia southern ariaes characteristics of the data for each mountains) appeared to be fairly similar in the plot and refers to geographical factors, and occurrence of peaks and dips. Therefore, because table 2 refers to biotic factors, detailed models of precipitation are being developed by others In the project, climate Step 2,--Developmeasures of recent increases variables were not included at this stage. or decreases in ring width for each plot relatLve Upon completion of the exploratory data to forecast values, analysis, i time series of ring widths was Time Series Analysis.--lo determine the recent constructed for each of the 44 remaining plots pattern of tree ring growth on each plot, an

Table 1,--Plotcharaereristies and geographical variables

ObservatLan Plot ELEV ASPECT l$iT LONG

lots eliminated from study because no red spruce was present. Table 2,--Plotcharacteristics and biseie variables

Observation DEADTREE TREEX

*plots eliminated from study because no red spruce was presant, autoregressive-integrated-moving average (~1~~)Eany a% the series were described satisfactorily time series model for each of ehe plots was first by an autoregressive scheme of order 2, developed, Kenda3-1 and others (1983) and occasionally with higher order moving average Vanciaere (1983) provide details of AF?If"rP, models (MA) terms. and the underlying assmptiocs . The MA terms improved the fit as measured by Time Series iYodeLing.--Variousyears sf the the diagnostics but did not materially affect the series indicated marked trends of tree ring forecasts. The autoregression [ARC2)] growth. To aecomodate trends, the series were pefficients were usually both positive with $1 I- differenced where necessary. Other approaches to 42 in the range 0.6 ts 0.9, indicating a carry- this problem are covered in the Discussion over from one growing sepon 50 the next, as section. In this data set, it was never would be expected. *&en 41 + (32 exceeded 0.9, necessary to difference more than once. A nonstationarity in the series was evident, and ary of the fitted models is presented in dsfferencing was performed. A Low-order K4 table 3. From the table, it can be seen that scheme usually gave an adequate description of ry of autcregressive-integrated-moving average rncldefs for each plot

Plot P30. Difference + BR i"kA kl a 2 0 No 2 0 No 2 0 No 2 0 No 2 0 No 2 (7?) Yes 0 (LS4,LO) Short series No 2 0 No 2 0 No 4 0 Short series No 2 0 Short series Yes 0 1 Short series Yes 2 0 Short series Yes (3) 0 Short series Yes 0 (4) Short series Yes 0 2 No 2 (8) No 2 0 Yes 0 2 lies 1 0 Yes 0 1 Yes 0 3 No 2 0 No 2 0 No 2 0 No 2 0 Yes 0 (2,3) Ye s 0 (1,2,6) No 2 C 6 1 Yes 0 (1,2,6) No 2 0 No 2 0 No 2 (lo?) Structural change in No 1 0 series? No 2 0 No 2 0 Yes 0 2 First seven terms deleted Yes 0 0 Yes 0 2 Short series Yes 0 1 Yes 0 (1,4,5,63 Yes 0 (2,5) Yes 0 (1,4) Short series No 2 (3,7?)

+ k indicates lags 1, 2, . . . , k (k) indicates lag k only (j ,k) Indicates lags j ,k only (k?) indicates lag k a possibility

AR =. autoregression coefficient MA = moving average the differenced series, It Ls well known that AR have involved fitting to 1464 (or 1974) and then schemes with a roo$ of the auxiliary equation fsretzsting, However, the risk of structural pear unity can often be well approximated by a changes in the series was such that the pure low-order MA scheme with a single difference. forecasts might misrepresent recent trends. Therefore the modeis are not very different ln Although our approach biases the residuals practLce despite their distinct theeretkeai somewhat towards zero, the method seemed to pr~pr+-igs provide s clearer picture of recent deve2spmenss, The only series that caused major problems was Because changes in ring width might be that for plot 3.09, where major increases in the considered in either ahsolute or percentage first 7 years were followed by steady declines, terns1 also considered were the indicators: After the data for the first 7 years were deleted, a satisfactory model was fltted, It can average of residuals proportional change be assumed thar the use of ring widch rather than - average ring width tncremental basal area was the cause of these nsnstatisnarity problems, aver the two forecast periods, These values are Measures of Recent Relative Change in Ring also listed in table 4 as PCT20 and PCT10. Width,--To assess recent relative increases or Assessment of Mean Change, - -One should note decreases in ring width, each series over the whether the residuaf ring widths rare below the periods was forecasted: expected value of zero for the perlods 1, 1965-1984, using 1964 as the forecast eonsldered, The results of one-tailed t-tests on origin, the data kn table 4 were as follows: 2, 1975-1984, using 1974 as tke forecast origin, "fhe residuals, the ddifference htween observed Adjusted and gredlcred va2ues, were then computed for each year in the period, The means and standard deviations of these residuals were computed for each plot (table 4). It should be noted that the models were fitted to the entire series, 1900-84, and forecasts were then generated from the forecast origin, A pure forecastina method would

Table 4.--Summarystatistics from tine series analysis

Observation OVL.ZEAI?I RES28 R2QSD PCT28 RESkO RIBSD PCTlO Table 4,--Sumvarystatistics from time series aaIysis--Continued

Observation OVMmJ RES20 R20SD PCT20 RESIO RlOSD PCTlO

+ OVlilEAF7 - overall mean of series. RES20 - residuabs from forecasts for last 20 years R20SB - standard deviation of RES20 values, PCT20 = RES20JOVMEAV. RESLO, RlOSB, PCTlO are defined similarly.

" Plots eliminated from study bemuse no red spruce was present.

The adjusted t-values were computed following the where TREEX is proportion of Live trees, and SBAX approach described by Cliff and Ord (1981) is the proportion of live basal area. The values modified to the one-sanrgle case. The adjustment of these variables are bfsted in table I. takes account of the positive spatial dependence The quadratic factors of Latitude and longitude among the data and may be written as: were included to allow a low-order trend surface analysis (Cliff and others 1975). However, tadj = t (1 - I), initial runs using only the geographical variables showed virtually no correlation between where I is defined in equation (1) under Step 4. any of these variables and the residual ring Evidently the 20-year resi&aal ring widths width measures; therefore they have not been hiwe a mean that is significantly less than zero, reported separately. A total of eight analyses while the null hypothesis of a zero mean is are reported in tables 5 through 8. For each of accepted for the 10-year values. Therefore a the residual ring width measures, the analysis drop is Indicated in average ring width in the was performed using both unweighted and wei&ted 1960" that has subsequently stabilized at that least squares (LS), The weights used were the Lower level, standard deviations given in table 4, In all eases, the significance level for a variable to Step 3,--Relaterecent increases and decreases leave or stay was set at 0.25. The residuals in ring width to geographical and biotic factors. from the regression analyses are given in tables 9 and LO. In this step, the changes in ring width were Interpretation of Regression Results.-- related to the various plot characteristics to Comparisons within and across tables 5 through 8 determine if there were any explanation for the show the following: changes . 1, The value of is in all cases somewhat Regression i9rialysis.--Eachof the four residual higher for weighted LS than for ring width measures was modeled using stepwise unweighted. A high standard deviation in regression with the following variables: the &$me series residuals shows an erratic geographical : latitude T longitude ring width pattern. Therefore the we igkt lng is useful because greater (LONG), (latitude)2 - UTS, (1ongitudel2 = emphasis given to the with more LOMG2, latitude * Longitude = MTLOKG, is plots elevation (ELEV), md aspect (coded as sine stable ring width development, Otherwise and eosine, SASP and GASP). the same variables were selected by the biotic: nmber of live trees (LIVETREE), stepwise procedures for both estimation nusnher of dead trees (DEADTREE), stand basal procedures, and the two sets of estimates area of live trees (SBALTVE), stand basal were broadly consistent for each of the area of dead trees (SBADEAB), and two derived four dependent variables. indices : 2. The proportional change indicators yield models with a higher degree of explanatory power than those based on absolute changes. Since average ring widths vary considerably between sites, use of the proportional change indecator seems preferable. Tzbble 5,--Regressionanaaysis lor the dependent variable 20-pear mean residual ring width (ItES20)

Sm of Mean

Mebe k 4 18813.450 2703,362 2,464 0.0608 Error 39 42758.529 1096,373 G total 43 53571.979 Root MSE 33.111517 R-square 0,2018 Dep mean -27,455909 Adj R-sq 6,I290 C ,V , -189.578

Intercept 1 76.597735 70,426505 1.002 0 3223 ELEV f -8.013309 0,010648 -8,287 0.2055 SBM I -69.517093 40,250901 - 1.727 0,0921 TREEX 1 91,734416 36.388501 2.521 6.0159 SBALPVE 1 -0.916871 0.442469 - 2.072 0.0449

Welghted least squares

Surn of Mean Source d f squares square I? value Prob>F

Mode k 4 800896 200224 4.859 0.0028 Error 3 9 1607051 41206.433 G total 4 3 2407947 Root HSE 202.994 R- square 0,3326 Dep mean -13.665677 AdJ R-SQ 0.2642 C.V. -14-85 -4-3

Parameter Standard T for H0:

Intercept 1 49.: 57126 81.765087 1.213 0.2324 ELEV 1 -0.017920 0.011910 - 1.505 0.1405 S BAX L -105.865 45.857356 - 2.309 0.0264 TREEX 1 135.208 39.969084 3.408 0.0015 SBALIVE 1 -1.201380 0.502896 - 2.389 0.0218

The regression analyses for geographical 4. The most important variable in almost all variables provided only very Littie cases was the tree index, TREEX, which is explanatory power. When the biotic probably an indicator of seaad health, and variables were also included, elevation strong, positive correlation is to be became important, and the residual ring expected. The other major biotic variable width became more negative as elevation was SBALIVE, but this appears with a increased. This suggests that the higher negative sign in the regression, SBADEAD elevation plots did worse than ayerage over and the stand basal area index (SBA) also the 10- and 20-year periods considered, appear on occasion, again with negative The only other geographical variables that signs in all eases. The interpretation of appeared in any models were LONG2 and SIN these effects is unclear, but these (aspect). The coeff Lcient on LON62 variables may relate to other biorie suggests a domward trend in the 10-year factors such as the age of the stand and change variables from east to west. Since the degree of competition, the plot Locations extended approxinrately northeast to southwest, this may reflect Overall, the weighted regressgons on the the influence of climatic factors. The proportional change indicators appear to give coefficient for SIN (aspect) indicates that a reasonslble explanation of the variations in the proportional ck~ange variable for ehe residual ring width. 10-year period is higher in plots with a southerly aspect. Again. this may reflect Step &.--Determine whether there is any spatial climatic effects. pattern to the values of excess or deficiency and Table 6.--Regressionanalysis for 10-pear mean resLdrtal ring w2dt.k (PXSMOj

Mode H 2 6777.608 3388,884 3.939 0,0272 Error 41 35269,460 860,231 6: total 43 42047,067 Root MSE 29.329690 W-square 0,1612 Dep mean 4,427273 Ad3 W-sq 0.1263 G.V. 662.4776

Parameter Standard T for HO:

Intercept i 814,995 404.97t 2.012 0.0508 SBAEIVE I -0.763085 0.303229 -2.517 0,0159 LONG 2 I -0.119851 0,059378 -1,951. 0.0579 -

Weighted least squares

Surn of Me an Source Of squares square F value Prob>E'

Model 2 301627 150813 &. 325 0,0198 Error 41 1429655 34869.627 C total 43 173l281 Root HSE 186.734 R-square 0.1742 Bep mean 7.603876 Abj R-sq 0.1339 C.V. 2455.775

Parameter Standard T for WO :

Intercept 1 1087,709 474.049 2.316 0.0257 SBALIVE 1 -0.852323 0.342500 -2.489 0.0178 LONG2 1 -0.156754 0.069541 - 2.254 8.0296

whether this can be accounted for by Under the null hypothesis (%) of no spatial and geographical biotic factors. autocorrelation (or independence), it may be In this section, the spatial methods used are shorn that: first described. Then the spatial analysis for the residual ring widths and for their residuals from the regression equations developed in step 3 is presented. The objective of the spatial analysis is to discover if there is any spatial Ord show tbe distribution of I pattern Ln the recent changes in rdng width, both Cliff and (1981) under Ha to be approximately normal, provided among the initial values and the residuals, from that n is not too small, For the configurations the regression equations. sf weights used and the rider sf plots available Spatial Nethsds.--Thefirst step in any spatial (n = 441, the normal approximation is analysis is to determine whether or not there is satisfactory. It should be noted that 1 is not any evidence of spatial pattern among the data, defined quite like a regular correlation given the plse locations, If the n plots have coefficient; in particular, the values tend to be observed values xi (l == 1, . . . , n), we set Zi = do the one would xi - S and use the spatial autocorrelation closer origin than expect. For this reason, the magnitudes of the standard statlscle: deviates

are often more useful than the values of P themselves. Prom Cliff and Ord f198P), the where So - 1 wlJ and the (wIi are a set of non- variance of I under Hg is: 1 J negative weights co be specifled, ~5thw~, = 0. Table 7,--Regressionmalysis for 20-year mean residual ring width divided by overall mean ring rsidrdi (PCT20)

Unweighted least squares

Sm of Mean Source dE squares square f value Prob>F

Model. 5 0,466992 0,0933518 2 - 592 0,0411 Error 3 8 1.369366 0,036036 C total 43 1.836359 Root MSE 0.189831 R- square 0,2543 Dep mean -0.107106 Adj R-sq 0,1562 C. Vu - 176,249

Parameter Standard T for HO:

Intercept 1 0,995842 0.583386 1,707 0,0959 ELEV 1 -.0000856071 .00006109282 - 2.401 0,1642 SBm 1 -1.074981 0,566563 -1.897 0,0654 SBAI)EPII3 1 -0,013608 0.011104 - 1.225 0.22753 TREEX I 0.579832 0,215236 2,694 0,0105 SBALIVE 1 -0.00389848 0.002933331 -1.329 0,1928

Weighted least squares

Sm of Hean Source df squares square F' value Prob3F

Model 5 35.672534 7,134587 6,669 0.0003 Error 38 44.671739 1.175572 C total 43 80.344273 Root MSE 1.084238 R-square 0.4440 Dep mean -0.073059 Adj R-sq 0.3708 C. V. - 1484.05

Parmeter Standard T for WO:

Intercept 1 1.311741 0.583284 2.249 0.0304 ELEV 1 -.0000828751 .00006361611 -1,303 0.2005 S BAX 1 - 1.582828 0.531610 -2.977 0.0050 s BAD EAD a -0.022~03 0.011483. - 1.969 0.0563 TREEX 1 0.800385 0.214337 3.734 0.0006 SBALIVE L -0.00426758 0.003139655 -1.359 0.1821;

Choice of Weights.--Given the irregular array of plot locations, the choice of weights for use in jl) is somewhat arbitrary. However, when the variables 1(1 are normally distributed, it is knom (Cliff and Qrd 1981) that I is the LaeaPIy most powerful test for alternatives of the form: where n - number of plots,

So = C C w,, 9 1 J where & and u2 are parmeters and W is s s, - C 1 I O when plots L and 3 are 3 .I ch~e,but wij -- 0 when they are distant. Table 8,--Regressionanalysis for LO-year mean residual ring width divided by overall mean ring width (PCTIO)

Unweighted least squares

Sm of Mean Source d1 sgkares sqicare -Gk~ue ProCrF

Model 7 0.678485 0,0969226 4,467 0,0012 Error 36 0,781208 8.021700 C total 43 2,459693 Root MSE 0,147318 R-square 0.4648 De-p mean 0.8216193. Adj R-sq 0,3607 6, V. 672.9362

Parme ter Standard 141 for 110: Variable

Intercept ELEV SAf P LON62 SBALZVE S EIm S BADEAD TREm

Weighted least squares

Sum of Mean Source cl f squares square F value P~OF

Hodel 7 27.779442 3.968492 5.605 0.0002 Error 36 25.489160 0.708032 G total 43 53.268603 Root MSE 0.841447 R-square 0.5215 Dep mean 0.035922 Adj R-sq 0.4285 C.V. 2342.437

Parameter Standard T for NO: Variable df estimate error

Intercept I 4.831085 2.305744 ELEV 1 -0000758574 .00005072042 SASP 1 -0.069365 0.033810 LONG2 1 -0.000484338 0.0003417928 SBALIVE b -0.00324435 0.002267685 SRdV 1 - 1.564668 0.436777 SEADEAD 1 -0.024423 0.009060183 TREEX I 0.554004 0.181281

Furthermore, given the rapid changes in local 2. W~J= 1, if plots i and j are first or topography that are possible in the study area, second nearest neighbors, otherwise it is reasonable to set qj = 8 when there are = 0. several plots between i and j. Given that (1) is These weights are not synunetrie, but this does being used primarily as an exploratory device, not cause any problems. In each case, a distance these guidelines may be incorporated into the set criterion was used to eliminate linkages across of weights by use of nearest neighbor linkages very long distances. The sets of neighbors are (Cliff arid others 1875). The exact specification arized in (B) of the Appendix. These weights of weights is not critical, Thus two sets of were used in a11 subsequent analyses. A program weights are considered: listing for a simple FORTW program to compute I I. wij = I, if plots i and j are nearest and the corresponding standard deviate is listed neighbors, otherwise in the Appendix (A). For the first nearest = 0. neighbor, Sg == 42, SI = 68, and S2 = 192. For Table 9.--Residuals from ordinary Least squares regression for ES20, RES10, PGT20, PalO, respectively

O"~servation mES2Q mES10 PGRES20 OGRES90 L 46,334 - 15.807 13,26047 - 8.04145 2 -21,678 -19 937 -0.13334 -0,1267S 3 -31.993 -15,635 -0,14916 -5.08203 4 -20,136 -40.225 -0.25344 -0,21953 5* 6 14,431. 10 682 0.lBPOB 0,12306 7 - 10,443 - 28.935 -0,57859 -0,13183 8 58.386 83,942 0,2113LI 8.17438 9 -12,145 - 15.249 -0.11850 -0,11586 10 -14.668 - 30.645 -0.$1789 -8,24663 11 - 17.193 -37.930 0.08341 0.00267 f 2 -11,397 - 3,404 0.04620 0,01085 2 3 23.945 42.668 0.14752 0,25374 14 55 , 349 30,302 0.27781 0.23123 15 - 10,846 24.604 -0,10033 0,05556 1.6 59,557 $3,661 0 . 371x9 0.14592 b 7 If.190 2.355 0.07027 -0.03010 18 12,921 - 18,585 0.09821 0.04559 19 -17.522 - 10.792 -0.29572 0.02309 20 - 39.395 -19.506 -0.13238 -0.02375 21 -80.534 -35.373 -0.27386 -0.035351 22" 23 -12.884 22.290 -0,05130 0.16258 24 -8.476 51.982 -0.03284 0.27436 25 -25.229 -112.730 -0.13405 -0,16915 26 22.956 -0.910 0.18552 0.02366 27 -2.885 - 14,081 - 0.04947 -0.13986 28 11.355 7,095 0.05162 0.04954 29 - 14.474 2,247 -0.07823 0.00537 3CIS" 3 1 -44.763 - 21, 355 -0.19854 -0.14849 32 -15.831 25,363 -6,08487 0.08936 3 3 -11,938 1,136 -0.07265 - 0.00901 34 51,827 15,167 0.31837 (a. 14705 3 5* 36 - 10,105 -11,982 -0.10263 -0.06974 3 7 16.486 8.212 0.04258 -0.01989 38 -6,060 -41,126 - 0.00886 -0.16374 39 9.809 - 32,049 6,03889 - 0.00258 40 117.441 -5.710 0.13757 - 0.01437 41 50.106 23,982 0.19884 0.01594 42 -15,341 - 22.285 -0.02459 -0.07543 43 44.717 46.526 8.26187 0.26913 44. 9.138 28,242 0.05684 0,07427 4 5 -61.570 -18.162 -0.47302 -0.20437 46 2.829 - 10.456 0,02660 -0,08615 47 -31.393 -16,830 -0.l.8230 -0.14232 48 30.003 19.321 0.24256 13.15529

* Plots eliminated from study because no red spruce was present,

first and second neighbors, Sg = 80, SI = 138, analyses shorn in tables 9 and 10. These values and S2 -: 670, are presented in table $2. The standard deviates The results for the four initial residual ring were computed using the same formulae as were width measures are s arized in table 11. The used to obtain table 11. Mare exact expressions spatial autocorrelation generally appears to be are given by Cliff and Ord (1981), but these are higher, based on the second nearest neighbor very tedious to use, and the differences in weights, Given that neigr~sring plots nay magnitude are generally slight. soaetimes have different aspects of soil InIrerpretation of Spatial Analyses .-The conditions, it can be assurred the set of weights results in cable 11 show clearly that pronounced based on the first and second order nearest spatial dependence exists, whichever measure of neighbors gives a more reliable indication of residual ring width is used. The first question spatial structure. that arises is whether such dependence can be Spatial autocorrelation coefficients were also ascribed to purely geographical effects. caLculated for the residuals from the regression Bowever, the geographical variables do not Table 10.-Residuals frcm weighted %east squares regressim for .lES20+ RESPO, PC1"2O, and PCIQLO, respectively

-21,521 - 27 "in., - 22.402 -45,974

" Plots eliminated from study because no red spruce was presented. provide any degree of explanatory power, and the major variables in these regressions related 430 spatial pattern of the residuals is essentially stand health. "Vhile this analysis indieaees that the same. The next questton is whether the the ring width changes are related to current biotic factors account for some or a11 of the stand health, the reasons for that current health spatial structure. status remain undetemined. "When tables 11 and 12 are compared, it may be seen that the level. of spatial autoeorre1ation has diminished In all eases. For the variable PCT20, the autocorrelation has become negative, The results of steps 1 through 4 suggest an but this may be due to the uncorrected effects of overall decrease in ring width increment amang autocorrelations among the explanatory variables. red spruce ir; the Great Smoky Mountains from 1965 One may generally conelude that the regression to 1975 or 1984, relative to time series analyses ilccomt for much, but not all, of the forecasts for those time periods. The magnitude spatial pattern Sorand in the residual ring width for these decreases appeared ko be related to values. H~wesler~one should recall that the elevation, factors of stand quality, and the Table 11.--Results of tests for spatial autocorreLation among original ring width residuals

variable' ~oefficient* Standard ~eviate*

RES I0 0.388 0.466 2.15 3.47

RES20 and RESlO refer to the residuals from the time series models for ring widths averaged over the forecast periods, 20 and 10 years, respectively. PCT20 and PCTI0 denote RES2O and RESlO divided by the overall mean ring width for the whole series.

* NN1 and MN2 refer to the sets of weights for the spatial autocorrelation coefficient based upon first and upon first- and seeond-order nearest neighbors, respectively.

Table 12.- -Results of tests for spatial autocorrelation among regression residuals+l/

Variable Coefficient Standard Deviate NNL NN2 NN1 NN2

Unweighted Regression

Weighted Regression

RES lO 0.266 0.268 1.51 2.05

'/see table 11 for definitions of variables, eoeff icients , and standard deviates. Regression residuals are listed in tables 8 and 9. relative decreases of first and second order 6, The inclusion of biotic variables in the nearest neighboring plocs, These findings should regression models serves to link the change be compared to those generated by other indicators to stand health, However, 4t does approaches to galn an understanding of the impact not resolve how the stands came to be in that of certajn key declslons In the stages sf the condiclon, The lack of any worehwhile analysis, correlations between the indicators and the Both Landau and others (1985) and Cook <1983) locatfonal varfables suggests that other recornended that annuah basal area growth factors may be at work in determining stand increment is preferable to rlng width as a health; furthermore, the high levels of measure of annual productl.vity. Use sf this spatial autocorrelation in the ring wtdth measure could have reduced the nonstationarity In change data indicated that such factors are several of the time series, One approach to time spatially concentrated, series that display marked trends is to transform 7. The extremely variable topography and the data (Cook 1987)- Instead, we Eellowed the locations of the sites suggest that ~urely usual ARIPcZi$ paradigm and differenced tke series spatial models (Cliff and 9rd 1981) are where there were marked trends; 20 of the 44 unlikely to be of direct value in this series actually required differencing. It would study. However, the potential exists for be useful to compare forecasts obtained from the worthwhile applfcations with more homogeneous untransformec8, but posslbly differenced, series clusters sf sites, to those obtained from transformed sertes. 8, Future analyses could include transfer Additionally, the average of the ring width of function modePs involving climatic the treeshores that met our incEusion criteria variables, ornee detailed models of these was used. Landau and others (1985) recommended variables have been developed. the use of trimled means, Specific circmstances of other data sets may deternine the advisability of one procedure over another as the best way to minimize heterogeneity, Cliff, A, D; Maggee, P,; Ork, J,K.; Basset, K,; The methods used in each of these steps are Bavies , W. 1975, Elements of spatial capable of further refinement, and severah serueture. Cambridge, England: Cambridge suggestions are included En the Recornendations University Press. 258 p, section, Nevertheless, the basic paradigm Cliff, A, B,; Brd, J,K, 1981, Spatial processes: represents a substantive approach to the Models and applications, London: Pion Press, evaluation of recent trends in the width of tree 266 p. rings, Cook, E, R. 3.987. The use and limitations 06 Results obtained from the steps taken in this dendrschronology in studying effects of air study should be compared with results generated pollution in forests. 1st: Procedings, of the from other approaches, Future studies might NATO advanced research workshop : Effects of include the use of basal area increment rather acid deposition on forest, wetlands, and than ring width as a measure of annual agricultural ecosystem; 3.985, May 12-17. productivity and may incorporate transfer Toronto, Canada. Springer-Veriag. 277-290. function models of climatic factors as well as Kendali, M, 6.; Stuart, A,; Ord, J.K. 1983. The intearvention analysis to filter out the effect sf advanced theory of statistics. Volume 3, 4th Important forest perturbations. Edition. New York: Macmillan. Landau, E.; Swlndei, B.F.; Thompson, S.R.; Van mc IONS Ryzln, R.J. 198%. FOEtAST review report. American Statistician. 39(4): 250-254, 2. Following Landau and others (19&5), it is Vandaele, W. 1983. Applied time serFes and Box- recornended that future studies should use Jenklns models. New York: Academic Press: incremental basal area rather than rlng 417 p. width. 2, The possibility of using trimed means rather than arithmetic means should be considered. However, the issue of how well measurements on five healthy trees reflect overall stand health requires further examination, 3. It Is iwortant to look for changes in ring FORTWV program for computing the spatial width or other indicators reiati-ve to what autocorrelation coefficient ml&t be expected. The forecasting approach used in this report is one way of excluding such trends, but others should be examined. The study 06 proportional changes seems preferable to that of absolute changes. BINENSION %(100), X(iOO), NOLOT(100), KNliLOO), 4. In future the series analyses, automated hT72("r06$, *bTl(loO>, rWT2(100, NGROSS(256) procedures might be used (AGTOBOX, developed READ, N Automatic Forecasting by David Reilly of READ, Sol, Sfl, S21, 502, SL2, 2322 Systems, knc,). SuLa=O. 0 5. Where known problems of fires, aphids, or DO LO 1-1 ,N infestations occur on particular plots, RW,MB, N1, N2, NREF intervention analysis should be used to NPmT (I)-NP filter out such effeets. ml(I>.-Na m2(1)-.292 NGROSS(NP)=NREF PRINT,TSBI-\,SDI,'SD~-'~,SD~ 10 CONTIrnE PRINT,' SPATIAL APC FOR FIRST h% IS \,SAG1 DO 15 1-I,N PRIN%,YPATIAL A/C FOR SEC0FlD NX IS \,SAC2 READ, SACI-(SAC1+8,o/(H-1>]/SD1 SUkfx= Stm*Au SAC~==(SAC~+IO/JN-%)),/SD2 x(r>=sxx PRINT,'S%D, DEVIATE FOR FIRST hi IS ',SAG1 i";Q?3Txw-!!Z PRINY "7% DEVXATE FOR SECOND IW TS ',%AC2 SllE3ZZ-SIr?i4Z4~0.0 STOP SRGl-SAC2-;O.O END XBm-SbmjN DO 20 1-1, N z(r>-x(r>-xBm SiiXZZ-.S"dMZZ+Z ( 1j **2 (B) Plot Nmber--first nearest neighbor--second SUMZ4-SUMZ4+Z{I)**4 nearest neighbor-- order of plot in listing of 2 8 CONTINE values (required inputs to vectors NPWT, m1, B2-N*SmZ&/(SUp.IZZ**2) NN2, -AND NGROSS) PRINT, %URTOSIS=\,82 DO 30 J-I ,N (continued) U-NNZ (J ) 25 13 37 14 KB=NN2 (9 ) 2-58 0 0 42 KG-MPLOT (J ) 13 37 36 12 IF (U.EQ.0) 60 TO 30 37 36 13 13 JA-NCROSSIU) 36 37 13 32 JC-.NGROSS(KC) 232 0 0 4.0 SACl=SACl+Z(JC>*Z(sA> IF (KB.EQ.0) GO TO 30 JB-NGROSS(KB) SAC2=SAC2+Z(JC>*Z(JB> 38 CONTINUE SAC2=SACL+SAC2 SACf=SAGl/SGMZZ sacz-sacz/suHzz SD1~N*((N*N-3*w+3~*SII-lu"*S21+3*SOZ*S01) SDI=SDI-B2*((N*N-E)*S1.1-2*N*S21+6*SOI*SOI) SDI-SDI/((N-I>*(N-2)*(N-3]*SO1*SO1) SDf=SDI-l.Od(N-1)**2 SDT=SQRT(SDl) SD~~=N*((N*N-~*M+~)*SI~-N*S~~+~*SO~*S~~) SB~=SD~-B~*((N*H-N)*S~~-~~.?S.N*S~~+6*S02*S02) SD~=.SD~/((N-~>*(N-~)~;-(N-~)*SO~*SO~) %B2-;SD2-I.O/(N-1)**2 SD2-SQRT(SDa) wit) = A(&) + B(t) + Gdt) + D(t) + e(t) wk1ere information from plots established in the Great A(t) - age factor(s) unique to each tree, Smoky Mountains by the National Park Service and B(t) - disturbances unique to each tree, the Tennessee Valley Authority was used to Get) - climarie effects eomon to all trees at determine anziial tree ring widths from core a site, samples of red spruce (Picea Rubens Saxg,) - D(t) - dismrkances colilixon to all trees at a red spruce care samples showed a significant site, and dependence of variance on the mean size of tree e(tj = random component. sings at 63 of 68 plots, At 9 of 48 plots, the variance has increased moue rapidly since 1943; It may be helpful to review some of the sf the others, 7 have shown a decrease since 5940 assunptions frequently made about W(t), The and 32 showed no change. The dependence of successive increments are assumed to be variance on mean of a measurement was interpreted identically distributed, For the benefit of in terms of "fractals," a term coined to denote certaLn analyses, the increments are further fractional dimension, The change in fractional assumed to be statistically independent, with the dimens ion over tine indicated an evolution of marginal distribution, e(t), Gaussian with zero factors that influeneed the dependence of mean, and constant variance, Such a time serdes variance on mean; these factors may have been is called a stationary Gaussian random walk or successional, climatic, or anthropogenic, aLL of Bromian motion. While no one working in which seemed to vary on about the same time dendrochronslogy seriously expects the Brownian scale. It was cone2uded that variance-mean assmption to be valid, the nature of statistics analysis may be an inexpensive and promising area often demands that it be assumed. of inquiry in dendroehronology. The concepts of randomness are context dependent, and the word may be used in a confusing manner, There are two broad, a1ternatLve definitions sf randomne s s : "predictable behavior, efficiently described by a Mortality of large forest areas in several statistical probability distribution'3arzd parts of the world has caused fear that the "haphazard behavior, governed by no known rules" concentration of anthropogenic eompsunds in the (Mandelbrot 1967). In addition, events in which atmosphere may be increasing. Acids and other the mean and variance are equal (Poisson oxidizing agents of hman origin, notably ozone, distributed) are often said to be random. have been detected In the atmosphere and are Predictable behavior is synonymous with probably capable of interfering with tree growth stochastic and differs from deterministic because and survival. However, there are no data its expected or average outcome is predictable, concerning the level of atmospheric pollution while the specific outcome of a trial lies within (Cook 1987, Kiester and others 1985). Therefore certain bounds defined by the variance. other possible influences and causes must be As stated above, statistical independence of considered, such as the effect of climate on tree successive increments is a well-known ring growth. Because climate and antbropsgenic simplifieatlon, The asswption of stationarity, effects are easily confounded, this report will however, has a special implication that is rarely focus on the analysis of change and not on questioned: the sample moments vary little from distinguishing between pollution and climate. sample to sample, provided the samples are large Annual tree rings in temperate regions provide sough. In our analysis, this assmptisn is a convenient record of a tree" growth history. relaxed, and the assmptisn is made that the Conrparison of tree rings over a geographical area variance is infinite or at least so large that it has frequently been used to determine cLiinritic may be treated as infinite. Thus the assmption changes; It is assmed that patterns cornon to of Gaussian marginal distribution is abandoned in all trees of the same species, similar age, and Eaavor of the Cauchy, permitting the Central Limit in the same soils should respond alike to weather Theorem to be invoked without assming constant condftlons that are basircaily unvarying (Creber variance (Mandelbrot 1969). Therefore the need 1877, Guiot and others 198%). to amend the other assmptions of independence and stationarity is alleviated (Berger and Mandelbrot 1963;. The assumption of infinite variance is It is cornonly ass--ed that the width, k;i'(t), of equivalent to asswing randomness thst is a tree ring laid down in year t is the linear sum predictable but haphazard; the long-term trend is of four systematic components and a random evident, but the short-term signal is component: unpredictable. Theory expounded by Plfandelbrot

R.A.J. Taylor is research scientist in the Department of Entomology, Ohio State University, Wooster, Ohio. (1968, 1963, 1969) related haphazard time series values vary according to no particulsc pattern; with power Laws and distributions with infinite however, it must be noted that the magnitude of varhnce. Vf(0) seems to increase with rime (figs. 1-8). Srnce Vt(8) is partly dependent on Mt(0), standardizing Vt(0) with Mt(0) may show a trend. Figures 1 through 8 also show the coefficient of Consider the rerips x(a;), t - 1,2,.... n, of variation I,/V~~B)/M~~QI~increasfng with time obsemations taken at equal intervals of tir-e or Figure 9 plots IsgV(t) against Lo@f(t) and space. Any particular series [x(t) is assuraed shows how the variance increases with respect to to be the realization of a process W(t) that will mean, The gradient dlogV/dlo@ - b is an be defined later; t is used as an indicator estimate of 13, This is easily demonstrated by variable at equal. intervals of time or space. the following argument, Define a reference mean Defining the following variables: 30 and a comparison mean Ms - sMg%(s>l . Now Vg - aOband Vs - absb - a(s~~]~- sD(a~o b ) = sbvO Evidently the variance is scale independent, and its gradient b is an intrinsic component, as expected from dlogV(k)/dlog(k). Taking logs: iogV, = iogV0 + blogfs), remernber that Vg is a variance corresponding to an V(x) is the variance of the series with M(x) arbitrarily chosen mean Mg, Vg is thus also the mean. Theory based on the usual arbitrary; Mg can be chosen such that Vg is I. interpretation of the Central Limit Theorem Thus logVs blag(s), which is very nearly permits one to assume that these parmeters - LogV(k) - &log(k), Although k and s are not quite estimate population values, However, in this synanpous, if multiples are chosen as values of analysis the assmption is not necessargly valid: then = describes the locus of the parameters simply represent the sample values s, TogV blog(i) + G variance at spatial/temporal inteavals i = and are therefore not asynaptotic to the 1,2 and is, except for identical to population values, C(x;k) is the ,..., C, alogK. Tkerefore B = b. spatial/temporal covariance across the data and logV(k) - The result can be obtained from an is related to the standard treatments of same empirical argtkment, there be a series of tirnelspace series data, including tree ring Lee samples taken along a transect AB. Divide AB analyses (Guiot and others 1982). Information is into NK intervals such that there are N groups of given in C(x;k) on the regular variations In the K intervals, Neither the K groups nor the N data at periods equal to k. The variogram is groups need to be contiguous, but for simplicity V(k) and provides information on the nonregular it 2s assmed that they are. The nmber of variation at lag k. entities in each of the NK intervals is counted, Consider V(k) as k varies: dividing V(k) - In the present case, the size of the tree ring 2fV(x)-G(x;k)) through by V(x) gives V(k) = increment is measured: xij where x is the size 26"(x)(l-R(k>], where R(k) is the serial and I l,2,..., M and j 1,2,..., R. Now compute correlation coefficient that takes values - - the mean Mi and variance Vi for each of the i - -IrR(k)sl; therefore V(k)=arV(x){l-r), Osrs2, and l,2,.. . ,N groups from OsTJ(k)S(x) Thus: . Mi Gxij/K Vi = X(xij -M~>2/f~-1). V(k) = O when serial correlation at lag k is Let the central interval of each group become 1, the center of mass for that group. The center of V(k) = 2V(x) when no serial correlation at lag mass has at least two parameters describing the k, and distribution of observations within the group: V(k) - 4V(x) when serial correlation at lag k Mi and Vi. is -I. If logV is plotted against lo#, empirically At any specific value of k, say k*, V(k) will they are related by the power law, V = &Ib, where give information on all variations not having a a and b are empirically estimable parameters cycle at k, (Taylor 1961, Perry 1981). Consider the To obtain information on a11 variations, V(x) arbitrary series xj* (j = l 2,. . . , that has is cornonly used, but we cannot say what mean and variance M* and v*; now compute the variation we have at t - i, only that it is over serial covariance ~*(k)having lag k, from the interval t i,i+l,i+2,. and so forth. - . . , c*(Ic) E x * - M*] [xj+k - M*]. would Ideally, the variance he partitioned in the From V* and Ci(d) the variance of increments is manner of V(k) but without Lhmitation. If k goes com uted: to zero, ail scales sf variation are included and !i V (k) = E(x * - xj+k*)2 - 2[~*- ~*(k)]. simultaneously V(k) disappears. To estimate Half ~*(k)is referred to as the variogram, and V(O), compute V(k) for bO and extrapolate back its computation is the first step in the to zero. To obtain the rate of approach of V(k) interpolation process known as kriging (Journel to the origin, plot logV(k) against Iogjk) and and Huijbregts 1978). The covariance term is determine the gradient, dlogV(k)/dLog(k) - l3 as C*(k) and is related to the systematic part of k->a. the ring increments, particularly growth at small Another approach to estimate the gradient of k. At larger k, Long-period variations such as V(O) is to replicate the generating process W(t): climatic cycles predominate. Filtering Wljt), W2(t), ....; V(O) can then be estimated at techniques (Guiot and others 1982) concentrate on any or all t. The estimates Vi(0) are all at the the structure of ~*(k) over varying periods of origin, so the gradient must be extracted from time. The total variance of the series V* them. Plotting the means and variances of contairis both the systematic and nonsystematic several series against time shows that the actual variance. MEAN ANNUAL TREE-RING WIDTH - SITE 1 (1850- 19814)

4

1850 1878 1896 'S'O 1330 '950 '9-C 19% YEAR

VARIANCE OF TREE-RING WIDTH

1850 1870 1890 1910 1936 1950 1370 :990 YEAR

COEFFIGiENT OF VARIATION OF IREE-WING WIDTH

FIgure 1.--Tiweseries of mean, variance, and coefficient: of variation of tree-ring increments From 1850 to 2984 at Tennessee Valley Authority site 1,

kfEAN ANNUAL TREE-RING WIDTH - SITE 23 {I850--- "i884)

YEAR

VARIANCE OF %-RQE--WING WIDTH

YEAR

COEFFICIENT OF VARIATION OF TREE-RING WIDTH

Figure 3,--Timeseries sf meaa, variance, and coefficient of variation of tree-ring increments from I850 to 1984 at Tennessee Valley Authority si te 23. MEAN ANNUAL TREE-RING WiBTX ---- SITE 36 (1850----- 4984) ?GO -j

YEAR

VARIANCE OF TREE-RING WIDTH

COEFFICIENT OF VARIATION OF TREE-RING WIDTH

'559 8 '890 9 $930 1950 IS79 '990 YEAR

Figure 4.--Timeseries of mean, variance, and coeffieiis~ztsf variation of tree-ring increments from 1850 to 1984 at Tennessee Valley Authority site 36. MEAN ANNUAL TREE-RING WIDTH - SfTE 307 (1901 - 1984)

tw 1910 1920 1930 1940 1950 1960 1970 1980 1990 YEAR

VARIANCE OF TREE-RING WIDTH

YEAR

COEFFICIENT OF VARIATION OF TREE-RING WIDTH 92 11 I- 10 5 5 09 > & 0 6 I- 07 Z 06

0, U.2 w 0 o4 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 YEAR

Figure 5.--Time series of mean, variance, and coefficient of variation of tree-ring increments from 2850 to 1984 at National Park Service site 307. MEAN ANNUAL TREE-RING WIDTH - SITE 310 (3901 - 1984)

YEAR

VARIANCE OF TREE-RING WIDTH

YEAR

COEFFICIENT OF VARIATION OF TREE-RING WIDTH

YEAR

Figure 6.--Time series of mean, variance, and coefficient of variaeion of tree-ring increments from 1850 to 1984 at National Park Service site 310. MEAN ANNUAL TREE-WING WIDTH - SlTE 316 (1901 - 1984)

3 in -4 -u 4

YEAR

VARIANCE OF TREE-RING '1PdIDTH

COEFFICIENT OF VARlATION OF TREE-RING WIDTH

Figure 7.--Timeseries sf mean, variance, aad coefficient of variation of tree-ring increments from 1850 to 1984 at National Park Service site 326, MEAN ANNUAL TREE-RING WIDTH - SITE 325 (490"i l"1984)

VARIANCE OF TREE-RING WIDTH

YEAE

COEFFICIENT OF VARIATION OF TREE-RING WIDTH

'900 '310 '920 1330 1940 1950 :960 1970 '980 1990 YEAR

Figure 8.--Timeseries of mean, variance, and coefficient of variation of tree-ring increments from 18.50 to 2984 at National. Park Service si ce 321. This study focuses primarily on the systematic nine sites showed no difference and seven showed variance comon to all trees at a site. a significant decrease since 1943, three sites Therefore the variance across all cores at a site had a nonsignificant regression, in each year was calculated to form the series Preliminary analyses with multivariate methods Vt. The comon systematic components at a site (principal components analysis, PCA) failed ta formed a baseline from which all other components identify any differences in regression gradient of variance were referenced. To find the eomon attributable to differences in the two data sets. systematic component, variance was plotted Variables ineluded in the PCA were annual against mean to standardize variance. The mean rainfall, mean annual maxim- and minimu was asswaned to be linearly related to the temperatures, xeric/mesic status, altitude, and baseline. stand basal area. Also to be noted is the change in the comon component, represented by the change in Vt relative to Kt. Changes between the relationship of Vt and Mt indicate changes in the normal Tree rings reflect the net effect of age, behavior of the process W(t). Changes result health, soil, and biotic conditions on tree from evolution in the process variance. If a growth. long-term change is anticipated, then evolution Age and health are unique aspects of each in the system process can be demonstrated by individual tree, but the environmental determining differences in the rates of change of canditions experienced by all trees in close variance before and after the reference point. proximity are considered similar or comon external factors. However, competition is one aspect of a tree's external environment that is considered unique. Therefore three influences of growth can be Of the plots established by the National Park established: Service (NPS) and the Tennessee Valley Authority 1. Age/growth---uniqueinternal factors, (TVA), only those with five or more red spruce 2. Competition---uniqueexternal factors, and trees in the sample were selected for analysis. 3, Environment---cowonexternal factors. With 2 cores (series) per tree, means and When samples are composed of mature and/or old variances for each year from the beginning of the trees, the age-dependent variance will be series were computed from 10 ring width relatively small and may even be missing in the estimates. Series ranged in length from 40 to younger trees. When very young and very old 135 years. These data yielded 20 bivariate trees are included in the sample, agelgrowth is series with length of approximately 84 years (NPS likely to have an impact only in the early part data), and 48 series with lengths varying from 40 of a series and variances are higher than normal. to 135 years (TVA data). The logarithm of Competition between trees in the sample may variance was then regressed on log mean with each help one to understand the dependence of variance point representing 1 year. on mean. For a given value of the mean, a high variance indicates a wider range of individual RESULTS responses. As the mean increases, the dependence of variance on the mean suggests that only some Figures 1 through 8 show the mean (M), variance elements of a sample are increasing, while others (V), and coefficient of variation (CV) against may be decreasing or increasing very little. If time at TVA sites 1, 9, 23, and 36 and NPS sites increases in tree ring width were all equal at a 307, 310, 316, and 321. Both the mean and site, the variance would be increasing variance were highly variable, and the plots of proportionately and b - 1. The mean value of b CV against time gave the strong impression that was 1.77, suggesting that at the majority of the variance was increasing with time. sites, when conditions were conducive to growth Furthermore, the detailed behavior differed increases, only some trees responded. A change greatly from site to site. Figures 9 through 16 in b over time was interpreted as a change in the show variance-mean regression plots for the same relative competitiveness of the trees at a site 8 sites, and table 1 shows the results of due to a change in the external conditions. regressions of 68 sites. Sixty-seven regressions were significant at probabilities of less than CONCUJSIONS 0.05. Only site 113 showed no dependence of variance on mean. Because the relationship between variance and To investigate the possibility that conditions mean is still uncertain, no definite conclusions in the post-World War I1 era were different from can be made about the influences on tree growth. prewar conditions, the data of sites with runs Because the nature of fractional dimension is longer than 80 years were split at 1943 and the still poorly understood, the theoretical variance-mean regressions repeated to test the treatment is not rigorous. Only since the hypothesis that the acceleration of variance with publication of Nandelbrot's book (1982) has there mean has changed. Table 2 shows that of the 28 been an increased awareness and interest in TVA data sets suitable for analysis, 8 showed fractals among scientists other than topologists. increases in dependence, 18 showed no change, and The ideas developed in this analysis were none showed a decrease in dependence. There were preliminary, but the reduction of the masses of two sites with a nonsignificant regression, tree core data to a single parameter, b, was a Interestingly, the NPS data showed the reverse new contribution to statistical methods in pattern: only one site showed a significant dendrochronology. However, the results of increase in variance-mean dependence since 1943, splitting the data into two sections were SITE 9

0 0 0 QOO D 0 0 0 06

2.00 1.50 1.2 1.4 1.6 1.6 ILO 2.2 2.4 2.6 LOG (MEAN) LOG (MEAN)

Figure 9.--Variance-nem plot (fog scale) of Figure 10. - -Variance-meanplot (log scale) of Tennessee Valley Authoriry site 1. Tennessee Valley Authority site 8. All regressions are significant at All regressions are significant at p<0. 01. This plot shows some pc0.01. This plot shows very good nonlinearity . regression.

SITE 36 SITE 23

83 1.8 1.9 Zk3 21 2.2 23 2.4 2.5 2A LOG

Figure Il . - -Variance-mean plot flog scale) of Figure 12.--Variance-mean plot (log scale) of Tennessee Valley Authority site 23. Tennessee Valley Authority site 36. All regressions are significant at All regressions are significant at p<0.01. This plot shows very good p<0.01. This plot shows very good regression. regression. 51 SITE m"7 SITE 310

Figure 13,- -Variance-mean plots (Log scale) of Figure 14.--Variance-me= plots {log scale) of National Park Service site 307. ALL National Park Service site 310. ALI regressions are significant at p

SITE 521

Figure 15.--Variance-mean plots ('dog-scale) of Figure 16.--Variance-mean plots {Log scale) of National Bark Service site 316. All NationaL Park Service site 321. All regressions are significant at p

Table 1.--Regressionanalyses of the log (variance] against Tog (mean) of tree ring increments+

Site Stzlndard Adjusted F- No. N Log(a) b error of b I?? ratio Significance

Tennessee Valley Authority Plots

wjs

Tennessee Valley Authority Plots *** Table 1. --Regressionanalyses of the Log (variance) against log (mean] of tree ring increments--Continued,

Site Standard Adjusted F - Xo. N Log(a) b error 06 b I$ ratio Significance

National Parks Service Plots

" A11 regressions are significant at P <0,0001,except where indjcated: N/S = not significant, * = P <0,05, ** = P <0.01, *** = P

Table 2.--Regressionanalyses of the Log(variance) against Log(mean) of tree ring increments before and after 1943+ - Site Bre/ Standard Adjust F- No. post N LogCa) b error of (b) It2 ratin Signif.

Tennessee Valley Authority Plots Table 2. - -Regression maEpses of the Iogjvariance) against iog(mean) of tree ring incremm ts before and after d943+- - Contimed

Site Prej Standard Adjust F- No. post N hg(a) b error of fS) It2 ratio Signif.

Tennessee Valley Authority Plots

National Park Service Plots Table 2. --Regression analyses sf the logevariance) against Log('mean) of tree ring increments before and after 1943'--continued

Site Tre/ Standard Adjust F- No. post N Log(a) b error of (b) It2 ratio Signif.

National Parks Service Plots

+~ll.regressions are significant at P <0.0001, except where indicated: N/S - not significant, * - P 4.05, ** = F <0.01, *** - P <0.001

LIT CSLTATIOMS pollution on forests. Internal Report of the National Vegetation Survey of NAPAP. New Orleans, ISI: U.S, Department of Agriculture, Berger, J.M.; Mandelbrot, B.B. 1963. A new model Forest Service, Southern Forest Experiment for error clustering in telephone circuits. Station. [unpublished]. IBM Journal of Research and Development. 7:224- Mandelbrot, B.B. 1960. The Pareto-Levy law and 236. the distribution of income. International Cook, E.R. 1987, The use and limitations sf Economic Review. 1: 79-106. dendrochronology in studying the effects of air Mandelbrot, B.B. 1963. The variation of certaln pollution on forests. In: Proceedings of the speculative prices. Journal of Business. 36: NATO advanced research workshop: Effects of 394-419. acid deposition on forest, wetlands, and Mandelbrot, B.B. 1967. The variation of some agricultural ecosystem; 1985, May 12-17. other speculative prices. Journal of Business. Toronto Canada. Springer-Verlag. 277-290. 40~393-413. Creber, E.T. 1977. Tree rings: a natural data- Mandelbrot, B.B. 1969. Long-run linearity, storage system, Biological Reviews. 52: 349- locally Gaussian process, H-spectra and 383. infinite variances. International Economic Guiot, V.; Berger, A.L.; Munaut, A.V. 1982. Review. 10: 82-111. Response functions. in: Hughes, M. M.; Kelly, Kandefbrot, B.B. 1982 The Fractal Geometry of F. M. [and others,1 , eds . Climate From Tree Nature, San Francisco: Freeman Publishing. Rings, Cambridge: Cadridge University Press: 461, p. 38-50 Perry, J.N. 1982. Taylor's power law for Journal, A.G.; Huijbregts, 6.3. 1978. Mining dependence of variance and mean in animal Geostatisties. London: Academic Press. 680 p. abundance. Journal of the Royal. Statistical Kiester, A.R.; Van Duesen, P.C.; Dell, T.R. 1985. Society. 30:(C) 254-263. Status of the concepts and methodologies used Taylor, L.R. 1963.. Aggregation, variance and the In the study of the effects of atmospheric mean. Nature. 189: 732-735. Red Spruee Tree Ring fysis Uslag a blmn Filter

Paul C. Van Deusen

pith date, and regional weathey data. National Ceatter Scrzice C1imati.e Diaision data frs~ A Kalman filter was applded to red spruce stations in the northern mountains of North (Picea rubens Sarg . ) tree ring data collected Carolina were s arized to provide total from the Great Smoky Mountains by the Tennessee rainfall and caverage tenrperature by month Valley Authority and the National Park Service. beginning in 1931. A new standardization method was developed that can be Justified wlth a model-based assmption. The variance of the standardized growth chronology appears to have increased in recent A basic goal of many tree ring studies is to years, The sensitivity of red spruce to climate produce a eonuaon chronology from a group of trees began to increase in the late 19609s and leveled to be representative of a site. It is probable off in the early 19802s, It is possible that that this chronology will demonstrate a eomon increasing climatic sensitivity and varlance are signal to which all trees in the area have related to balsam wooly adelgld activity in these responded. In order to aplify the coqon stands . signal, an attempt is made to eliminate individual tree signals that are unrelated to the comon signal. The age-related biological growth signal can be removed by a nder of procedures Tree ring data provide one of the few eategoricaPly referred to as standardizatlon. historical records for scientists to assess the Graybill (1982) presents methods here a growth impact of atmospheric deposition influences on model is individually fit to each tree ring forests (ADIF). However, to adequately assess series. this iqact, historical information on weather Graybill (1982) presents a hypothetical and pollution levels are also needed. Although breakdovan of t.he raw ring width for a single tree past weather records are mailable from weather at time t, R(t), as follows: stations, deposition levels can only be inferred from proxy variables such as coal consmption of nearby power . Ky analysis will be limited to examining the trends in the tree ring series where Ct is the rnacvoclilnatic signal comon to and attempting to explain these trends wlth all trees ; average monthly temperature and total Bt is the biological growth trend - a precipitation records. The Kalman filter is function of tree age; shown to be a useful tool for this type of Dl, is a disturbance signal that is unique analysis. to the individual ; DZt is a disturbance signal comon to most or all individuals - -possibly caused by fire, insects, or pollution; and The National Park Service (NPS) and the et accounts for random disturbance. Tennessee Valley Authority (WA) had collected data from plots located in the Great Smoky In order to maximize the macrosignals (C and Noun ta ins. Spatial relationships, current D2), it is necessary to recognize and remove the diameter, and species were recorded for each tree microsignals (B and Dl) as much as possible (Cook on each plot, A few dominant trees were selected 1987). An index is formed as Eollswes: at each NPS plot from trbieh two increment cores were taken, but pith date was not recorded; tree rings were dated back to 1900. Dominant trees on TVA plots were not cored, but pith date was where Y(t) is the model-based prediction of R(t). recorded; unfortunately tree ring widths were This produces a new index series with an only wallable back to 1850. Information from expeetation of I, a more homogeneous variance, qproximately 200 red spruce trees were available and a smaller first order autocorrelation than in both data sets, h data set from a site on the original series (Frltts 1976). GrafliLI Clingman" Dome (North Carolina) consisting of 38 (1982) presents negative exponentials and cores on 3.9 trees was also utilized in this orthogonal pol~omials as potential prediction study. Ed Cook of the Launont-Doberty Tree Ring models. Laboratory collected and cross-dated this data Cook (1987) discusses the use of splines to set. The TVA and NPS tree ring data were replace Graybill's nodels. There is a processed at Oak Ridge National Laboratories. possibility that the disturbance signal (Dl) as Generally, only data that would be commonly well as the B-signal may be removed with the axaaLLable in a dendrochronological study were spline approach, and more user interaction and used; these data included ring width, elevation, expert opinion are required. Warren (1980)

Paul Van lleusen is a mathematical statistician with the Southern Forest Experiment Station, Forest $emice-USDA, New Orleans, Louisiana 70113. presents an alternate model-based approach that cou% also be used to remove the D1 signal, but it also requires much user interaction, A system for updating and predicting is presented by Kabman (1960) based on a linear dynamic model, These models are genera%lzations that can generate any of the class of models (Box and Jenkrns 1976), standard muLtfp&e a method was ssughe for thls study chac regression models, and regression models with required little subjectivity and could be used to time varying parameters (Harvey 1981)- Applied automatically process many hundreds sf cores, to dendrockronoiogy, the KaTman falter provides a The gethod begins with a standard sigmokdal model means sf simultaneously reducing a number of for dia~eterat same age, D(A): series to a single chronology and generating climate-based predictions. Furthermore, the elfmate parameters can be allowed to vary ovsr time to provi.de a test of the uniformitarian where b is the asymptote parameter, and principle that conditions In the past are similar k is the shape parameter, to the present, The uniformitarian principle Is Differentiating model (3) with respect to age the Eundwenea% justification for the use of gives a diameter growth nebel: dendrochronoLsgy to infer past csmd%tions, The Kakman fdlter can be derived from Bayesian cheory {Harrison and Stevens 1976; Meinhold and Slngpumal%a 9983) or with least squares methods presented by Duncan and Horn (197%), Tke equations needed to implement the Kalman filter are presented below, and the reader is referred Model (4) is appropriate for radial increment to the citations for the theoretical development, data and is similar to standard The basic difference between the Kainan filter dendrochrsnakogicaE methods, and usual regression models is that the Assuming model (41, two steps are required to parameters are allowed to -vary over time, The remove tFle age-relate6 trend from the data, relationship between the vector of observed First take the natural Log of model (41, giving: standardized ring widths at time t (Yt) and the parameters (at) hs called the observation or logjR(A)] = constant -M Q 5> measurement equation:

Then take the first differences of model (3, giving : where the matrix Ft is fixed and of order ntxp, at is a pxl vector of underlying state parameters, and sat is an ntxb vector of residuals with zero expectation and variance matrix Vt, Thus a simple kransfcrmation remoa9es the age- related trend from the tree ring series, Result The state variable, at, evolves over time (4) can be justifled intuitively by viewing it as according to a first order Marksv process as a relative growth rate. Taking the Isg of R(%) defined by the transition or system equation: puts it on a relative scale, and the first difference is just a nmerical first derivative that can be viewed as a growth rate, This transformation can be quickly performed without where Gt is a fixed pxp matrix, and wt is a pxl sophisticated software or user interaction, an vector of residuals with zero expectatson and important advantage for Large data sets. variance matrix Wt, Plotting the data before and after The error terms '"st and wt are assmed to be transformation indicated thak the new series was independent white noise series, The quantities stationary and that the age-related trend had tkat must be known include the matrices (Ft and been removed as expected (fig. 1)- Figures I and Ct) that premultiply the state variables (at and 2 show hew the transformation creates similar at-1). Zhe matrices Ft and Gt correspond to series from a young tree and an old tree that independent varhables 7Ln regression theory. or, looked quite different before transformation, when dealing with rocket traj eetsries , they come Notice that 1937, 1969, and 1981 are all Isw on fri m well-deflned physical laws. The more the transformed series, esmylex problem comes from the need to know the The analysis can then proceed on the variance matrices (Idt and Vt), because in many transformed data by assuming tkat the new series statistical applications this will require some is composed of the EeLZowing signals: user subje~tivity . The equations needed to estimate the state variables can be divided into three parts: prediction equations, updating equations, and where Ct, Dlt, B2t, and et are defined as Zr smoothing equations, Let at-1 denote the optimal model (1). estimator of at-1 based on all information up to The age-related signal is WOW removed, Two and including Yt-l. The covariance matrix of macrosignal terms that are of interest remain (G at-% - at-1 will be Pt-l. The prediction and D2), as well as two uninteresting microsignal equations for at and the associated covariance terms (Dl and e) that will be treated as noise. matrix given at-1 and Pt-l are: 0 1915 20 25 30 35 48 45 5Q 55 66 65 70 "7 50 85 YEAR

1915 20 25 SO 35 40 45 50 55 60 65 70 75 80 85 YEAR

Figure 1.--Ring widths of a relatively young red spruce in raw (upper graph) and standardized (Lower graph) farm for a young tree, First differences of t-he natural log were used Co produce the Lower standardized series.

atit-l = Gtat-l , and estimate at time t-P and a weighted average of P,/,-l = GtP,..lG," VW, the prediction errors (Et), At any time, at is an optimal estimate, given Wlen Yt becomes available, the updating all previous information; but only the estimate equations for the estimate of at and the at time T (the final period) contains all of the associated covariance matrix are: information available. Given all. information, the optimal solution for any time t is referred

at = + Ptlt. lFt'z~-;E,, and (&a> to as signal extraction or smoothing. Smoothing begins with the solutions ae time T and Pt - P,/t-1 - ~,/t-lF,"t- FtP,/,-l * (4b) where recursively goes backwards to time 1. This yields the optimal smoothed estimates of the Et = Yt - Ftatjt-1, Zt = FtPt/t-lFt -I- Vt , and, for computational state variables with associated covariance savings, matrices as follows: 2t-l - vt-l - vt-'I7 [F~'v~-'F~ -+ Pt/t-I-'] -fFt'Vt-'.

The estimate of at in (4a) is the sum of its I .I4

0.86

0.57

8-29

0

- 0.29 - 0.57 - 0.86

- 1.14 1900 05 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 YEAR

Figure 2.--Ring widths of an older red spruce in raw (upper graph) and standardized (lower graph) Eorm for an old tree. First differences of the natural log were used to produce the lower standardized series. where aT/T - a~ for the starting value on the chronology can be fomed simultaneously with its state variables, climate predictor, and the second is an PT/T = PT for the covariance starting application to dendroclimatology. For art values and additional application, readers should see Jones

pt* = PtGt+lP ~t+l/t-l (1980) where various models are fit to drought data reconstructed from tree rings,

The Kalman filter provides a complete system for prediction, updating, and smoothing that The Kafman filter was applied to the red spruce should appeal to the dendrochronologist, In data collected by the NPS and WA. The objective particular, the usual procedure of climate was to simultaneously form a single chronology prediction from a single chronology formed from and its climate-based prediction. The data were many standardized tree ring series could be first standardized by taking the first difference refined. Two applications of the Kalraan filter of the logarithms. The following Kalman filter will be presented. The first shows how a was then applied: able 2. The climate variable used was a Linear combinetion of monthly rainfaL2 and temperature averages. The weight below is the va;ue Transition equation alt - Cta2,t-l+ wit (gb) nu2 tip2i ed by the corresponding man chly average @2t = a2.t-l + w2t (Ac) L-o creace this combined c1imat.e variable. Zlhe principle eomponects method was used to create 'rzFtt?~sa this 2i;tea~ ~0~5iridticilr. Sepcernber ere~"ipesatiire Yt Is the ntxl vector of observed Cree ring and Cctsber rainfall have the largest weights. data at time t, Ft is a matrix with two calunns of length nt with the first column being 1's and Variable Month Weight the second O's, vt, wit, and wzt are random errors, Ce is the climate variable, Temperature 3 alt is the value of the chronology at time F t, and M a2t is the climate effect at rime t. A M The covariance matrix Vt was defined to he 02t~twhere 02t was estimated from vector Yt as C(Ylt -yt)/[nt-l j , and It was an identity matrix of order nt. in other words, the trees were treated a prlori as independent. The covariance matrix Wt was defined as

Rainfall

Thus the state variables were assumed to be a priori independent. The elements on the diagonal were chosen to allow the state variables to vary enough to respond to a trend, but not enough to absorb random fluctuations, Since alt might approximate the average of the vector Yt, nt-l was chosen for the upper diagonal, and the factor 0.01 was used in the lower diagonal to prevent qt from fluctuating in response to random disturbances. The results were robust to changes in the W and V matrices, which lends credence to these values. Data from NPS plots dates back to 1900. The above algorithm was applied to the data from 1901 L960's, but did poorly before this. The corresponding plot of the parameter (the climate through 1983, because taking first differences effect) that multiplies climate also showed an eliminates the first observation. The climate increase over time. This indicated that the data were available from 1931 through so 1983, trees were becoming mare responsive to climate in the filter was modified during the "preclimate" the 1960's than they were previously. The reason period (1901-30) by setting wzt and Ct to zero. for the increased sensitivity to climate cannot be determined from this study. One might Results of Application I speculate that this was caused by thinning in the stands as a result of insect damage to the fir Climate lagged by 1 year was found to best component. L?nether this is related to pollution predict the chronology or time trend in the data. has not been determined. The cli_ntate variable used was a Linear coniGination of all the monthly rainfall and Application 2 temperature data for a year, as described in table I. The KaZ~an filter can be applied to Figure 3 presents the chronologies for the simultaneous prediction of past climate and the three data sets. The NPS and Clingman's Done single common chronology. This traditionally chronologies were very similar, particularly over involves averaging the individual series together the past 18 years. Variation in the standardized as a first step to form the single chronology. series had a tendency to increase in the more The climate prediction is then a separate step recent years. This increase was most evident in that takes place without the complete information the Tk7A chronology, but there was no suggestion contained in the original series. for the cause of increases. h Kalman filter can be formulated to handle The time trend predictions based on lagged these steps simultaneously. This incorporates climate and the climate effect variable a2~with the full information contained in the data while 95 percent confidence intervals are plotted in automatically providing a prediction system for figures 4, 5, and 6. Lagged climate predicted past climate with associated prediction the chrono~ogies accurately beginning in the intervals. Although the simple solution given CLINGMAN" SOME RED SPRUCE -TIME TREND

-0.40LLLLL1 1900 05 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 TENNESSEE VALLEY AUTHORITY RED SPRUCE DATA-

190005 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 YEAR

Figure 3.--Standardized chronologies from the Clingman's Dome, National Park Service, and Tennessee Valley Authority data sets. First differences of the log transform were used to standardize each tree ring series. A Kalm filter was used to produce 'the chronology as described in equations (6a), (db), and (6c). CLINGMAN*S DOME RED SPRUCE DATA LAGGED-

1900 05 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 CLINGMAN'S SOME RED SPRUCE DATA LAGGEtSs- .-'.--

I 8 8 I I

;'QO * I

-.-- -- 1930 3s 40 45 50 55 60 65 70 75 80 85 YEAR

Figure 4.- -Time trend predictions: A, Clingman's dome chronology (soiid line) and its climate based prediction ('dashed Line) using the Kalolan filter described by equations (&a), (6b), and j6c) ; B, the trend in the climate parameter given in equation (6a). The climate variable is a principLe component (linear combinacion) of monthly average rainfall and temperature variables. NATIONAL PARK SERVICE RED SPRUCE DATA LAGGED-CLOMBTE BASED PREDICTION

A

NATIONAL PARK SERVICE RED SPRUCE 7.5E-92 DATA LC1GGED-CLIMATE EFFECT AND I I

95 9'0 CONFIDENCE INTERVAL 88 II

2.OE-02 1.6E-02 1.2E-02 7.8E-03 3.9E-03 -5.OE - 05 -4.OE- 03 1930 35 40 45 50 55 60 65 70 YEAR

Figure 5,--Time trend predictions: A, National Park Service chronology (solid line) and its climate based prediction (dashed line) using the Kalman filter described by equations (6a), (6b), and (Sc); B, the trend in the climate parameter with 95 percent confidence intervals, given in equation (6a). The climate variable is a. principle component (linear combination) of monthly average rainfall and temperature variables. TENNESSEE VALLEY AUTHBRIW RED SPRUCE 0.30 DATA LAGGED-CLIMATE BASED PREDICTION

TENNESSEE VALLEY AUTHORITY RED SPRUCE 0.055 DATA LAGGED CLIMATE EFFECT AND 95 O/o - ,#*- - -- # I CONFIDENCE INTERVAL I I

- 0.009 I930 35 40 45 5C 5' 6t bE 70 75 33 85 YEAR

Figure 6.- -Time trend predlctinrs A, ier~rlessc-c3''a'Ler- ACA~JIOL11) C-!I~L-I:C: ?hT (solAd Lrne) 317d I~Sci,mare based ~rt'il~c~~i?;:(das!ii"~? i~nt--i LISLIJ~~

the KaLmar~ f~lterdescrrbed 5v eqzrations ioa! , 16b), 3x3 ii C-), B. the trend in the cllrnat-e pavmerer gll17eri Iri ~qirat~cnri63 Trlc? climate variable is a principle conlpone;:? il lnear co::lii~i;3ri01l) OT nmnt-hLy average ramfa: 1 a120 reli:perar1rre ~~rinb1e.s below does not achieve the potential of the predicted poorly (fig. 7)- The values before 1931 method, it was chosen to emphasize some important were predictions generated by the modal, Figure points, Consider the following EarwuLatLon: 7 shows the climate parameter a2 trendlng over time with a 95 percent confidence interval. The parameter was tending toward zero as the i7af confidence interval expanded, which is not Obse~~atlon surprisinq given the paor predirtinns etqu~iufiii Althc*~gh the model. presented in equations (723) through j7d) may not be ideal. this demonstrates how one could use the Kalman filter for predicting past climate. (7e: @mGmSION

(7d) The focus of this paper was to apply the Kalman filter to the study of tree rings. The Kalman filter provides a natural way of handling many of the problems that dendrochronslogists encounter. tihere Yc is the ntxl vector of standardized rir&g widths at time t, Ct is a qxl vector of observed climate GL~NGMAN" SOME DATA-PREDICTED VS ACTUAL. CLIMATE variables, $8- jt is a vector of lk sf length nt, Ot is a vector of 0" of Length nt,

Ut is the mean of the vector Yt, a-jt is the value of the chronology at time t , a2t is the climate parameter at time t, and vlt3 ~2~,wlr, ~2~ are random errors.

Equations (7a) through (7d) define a system that handles the dendrocLimatoZogical problems of forming a single chronology for a site providing a means of predicting past climate, and testing the uniformitarian principle. Some details on implementing the above system must be noted. The iterative process is started at time T rather than time 1, since past climate predictions are needed. Suppose that the climate CLINGMAN'S DOME CLIMATE PARAMETER TREND variables are available from time T to time t*, l2r where 1 < t* < T. During this interval of known climate (INT,) the parameter a2 is estimated. Beyond IMT,, equation (7b) is eliminated and the Kalman filter is allowed to generate new values of a2 back to time 1 (the earliest time for which tree data are available) and simultaneously produce the chronology and climate predictions as aztYt+p Furthermore, the trend can he examined In the a2 parameter over INT, to see if the uniformitarian principle is valid, although the Kalman filter will tend to move a2 along the established trend making the uniformitarian assmption less important.

Results of Application 2

Starting values must be supplied to the system in (7a) through (7d) for the parameter vector and Figure 7.--Climate prediction: A, preciietion the variance matrices Vt and Wt; Vt was assumed (dashed line) of pest climate obtained diagonal with -7ariances estimated from the from tree ring data using the Kalman vectors Yt as in exampie I. The last diagonal f iL ter described by equations (723) element in Vt is the vartance of v2t and was through (7d). The solid line is known estimated from the variance in the knewn climate climate, which is available back to data. T"ne matrix Wt was also assumed diagonal 1431 ; B , the climate parameter trend and chosen similarly to example 1. and 95 percent eonf idence interval. A principle component of climate variables was This plot indicates that the used for Ct. A starting value of zero was used uniformitarian principle does not hold for the chronology parameter al, and a2 was here, since the parameter is tending started at 10. Unfortunately, climate was toward zero. The flkrst application ip.;.*olsred prediction of an acid depesition on forest, wetlands, and average chronology with climate incorporated in agricu1tural ecosystem; 1985, May 12-17. the process. Tke parameter associated with the Toronto, Canada. Springer-Verlag. 277-290 climate was allowed to vary over time. The Duncan, B. B.; Horn, S.D. 1972. Linear dynamic rliwate gararnerer P'nllowed s sipold clzr-ae r-k.3~ recursive estimation from rhe viewpoint nf implied an increasing sensitivity to climate with regression analysis, Journal of American time. One might speculate that Insect-caused Statistical Association. 67(349): 815-821. thinning sf the fir component accounts for this Fritts, W.C. 1976. Tree rings and climate. phenomenon, London: Academic Press. 567 p. The second application gave same insight as to Graybill, D, A. 1982. Chronology development and how the method can be applied to more typical analysis, b:Climate from tree rings., M. K, dendrochronologieaL needs, i.e-, gredlcting the Hughes, P. M. Kelly, J. R. Pilcker and V. C. past value of climate from tree rings, It was Lamarche Jr., eds. Cambridge University Press, previously indicated in application L that the Cambridge: 21-28. data set employed was insensitive to climate in Harrison, P. J, and C. F. Stevens. 1476. the past and thus predictions were poor. Baesian Forecasting (with discussion). J Royal Although the filter was not extremely Stat Soe B 38: 255-247. sophisti,cated, it showed how one might proceed Harvey, A, C. 1981, Time series models. Oxford: with such a problem. Future mean tree ring Philip Allan Publishers Limited. 229 p. values were used to predict current climate while Mornbeck, J. W. and R. B. Smith. 1985, simultaneously developing the mean chronology. Documentation of red spruce growth decline. The trend in the climate parameter could also be Canadian Journal of Forest Research. 15: 1199- inspected as an indication of the validity of the 1201. uniformitarian principle. Jones, R. H. 4380. Naximwn liklihood fitting of models to time series with missing observations. Technometrlcs 23: 389-395. Kalman, R.E. 1960. A new approach to linear filtering and prediction problems. Trans. ASME. LIT CITD J, Basic Engr. 82: 34-45. Kiester, A. R., Van Deusen, P.C. ; Dell, T.R. Adams, H. S.; Stephenson, S.L.; Blasing T.J.; 1985. Status of the concepts and Duvick, D.N. 1985, Growth-trend declines of methodologies used in the study of the effects spruce and fir in mid-appalachian subalpine of atmospheric deposition on forests. Internal forests. Environmental and Experimental report for the National Vegetation Survey of Botany. 25(4):315-325. NAPAP, Southern Forest Experiment Station, 701 Box, 6. E. P. ; Jenkins, G.M. 1976. Time series Loyola Ave. New Orleans, La 70113. analysis: Forecasting and control. Holden- Meinhold, R. J.; Singpurwalla, N, D. 1983. Day, Inc. 575 p. Understanding the Kalman Filter, herican Cook, E. R. 1987. The use and limitations of Statistician 37: 123-127. dendrochronology in studying the effects of air Warren, W. 6. 1980. On removing the growth trend pollution on forests. In: Proceedings of the from dendrochronological data. Tree - Ring NATO advanced research workshop: Effects of Bulletin 40: 35-44. Van Deusen, Paul C., editor. 1988. Analyses of Great Smoky Mountain Red Spruce Tree Ring Data. Gen. Tech. Rep, SO-69. Kew Orleans, LA: U.S. Department of Agricu~ture,Forest Service, Southern Forest Experiment Station. 67 p.

Four different analyses of red spruce tree ring data from the Great Smoky Mountains are presented along with a description of the sprucelfir ecosystem. The analyses use several techniques including spatial analysis, fractals, spline detrending. and the KaLman filter.