## Info

Years A.D.

FIGURE 10.1 I Average age of the core segments used to produce a 1500-yr composite chronology from trees in northern Fennoscandia. Samples representing the recent period are generally longest because samples are usually selected from the oldest living trees, whereas older samples (dead tree stumps) may be from trees of any age (Briffa et al., 1992a).

RING WIDTHS

Raw Data

RING WIDTHS

Raw Data

600 800 1000 1200 Regional-Curve Standardization

1400 1600 1800

600 800 1000 1200 Regional-Curve Standardization

1400 1600 1800

1000

1000

### 1200 1400 1600 1800 Years A. D.

FIGURE 10.12 Ring-width data from trees in northern Fennoscandia showing (a) the mean indices without any standardization, (b) indices derived from standardization using the regional curve standardization, or (c) indices derived from standardization using a cubic-smoothing spline function (see text for discussion) (Briffa etai, 1992a).

FIGURE 10.13 Regional standardization curve (RCS) of tree-ring samples from northern Fennoscandia based on a least squares fit to the mean values of all series, after they were aligned according to their biological age (Briffa et al„ 1992a). Ring-width data commonly has a more pronounced growth function than maximum latewood density data.

Tree Age (Years)

FIGURE 10.13 Regional standardization curve (RCS) of tree-ring samples from northern Fennoscandia based on a least squares fit to the mean values of all series, after they were aligned according to their biological age (Briffa et al„ 1992a). Ring-width data commonly has a more pronounced growth function than maximum latewood density data.

common to all samples. The resulting "regional curve" provided a target for deriving a mean growth function, which could be applied to all of the individual core segments regardless of length (Fig. 10.13). Averaging together the core segments, standardized in this way by the regional curve, produced the record shown in Fig. 10.12b. This has far more low frequency information than the record produced from individually standardized cores (Fig. 10.12c) and retains many of the characteristics seen in the original data (Fig. 10.12a). From this series, low growth from the late 1500s to the early 1800s is clearly seen, corresponding to other European records that record a "Little Ice Age" during this interval. Also seen is a period of enhanced growth from A.D. -950-1100, during a period that Lamb (1965) characterized as the "Medieval Warm Epoch." It is apparent from a comparison of Figs. 10.12b and 10.12c that any conclusions drawn about which were the warmest or coldest years and decades of the past can be greatly altered by the standardization procedure employed. All of the high frequency variance of Fig. 10.12c is still represented in the record produced by regional curve standardization but potentially important climatic information at lower frequencies is also retained. The problem of extracting low frequency climatic information from long composite records made up of many individual short segments is addressed explicitly by Briffa et al. (1996) and Cook et al. (1995) who refer to this as the "segment length curse"! Although it is of particular concern in dendroclimatology, it is in fact an important problem in all long-term paleoclimatic reconstructions that utilize limited duration records to build up a longer composite series (e.g., historical data).

### 10.2.4 Calibration of Tree-Ring Data

Once a master chronology of standardized ring-width indices has been obtained, the next step is to develop a model relating variations in these indices to variations in climatic data. This process is known as calibration, whereby a statistical procedure is used to find the optimum solution for converting growth measurements into climatic estimates. If an equation can be developed that accurately describes instrumentally observed climatic variability in terms of tree growth over the same interval, then paleoclimatic reconstructions can be made using only the tree-ring data. In this section, a brief summary of the methods used in tree-ring calibration is given. For a more exhaustive treatment of the statistics involved, with examples of how they have been used, the reader is referred to Hughes et al. (1982) and Fritts et al. (1990).

The first step in calibration is selection of the climatic parameters that primarily control tree growth. This procedure, known as response function analysis, involves regression of tree-ring data (the predictand) against monthly climatic data (the predictors, usually temperature and precipitation) to identify which months, or combination of months, are most highly correlated with tree growth. Usually months during and prior to the growing season are selected but the relationship between tree growth in year tQ, 14 may also be examined as tree growth in year tQ is influenced by conditions in the preceding year. If a sufficiently long set of climatic data is available, the analysis may be repeated for two separate intervals to determine if the relationships are similar in both periods (Fig. 10.14). This then leads to selection of the

FIGURE 10.14 Results of a response function analysis to determine the pattern of growth response in Scots pine (from northern Fennoscandia) to temperature in the months September (in year 11) to October in year ^.Values plotted are coefficients from multiple regression analysis of tree-ring maximum latewood density (left) and ring widths (right) in relation to instrumentally recorded temperatures in the region.Two periods were used for the analysis to examine the stability of the relationships: 1876—1925 (dotted lines) and 1926-1975 (solid line). In addition to the monthly climate variables, the ring width values from the two preceding years (t, and 12) are also included to assess the importance of biological preconditioning of growth in one summer by conditions in preceding years.The analysis shows that maximum latewood density is increased by warm conditions from April-August of year t0, and a similar, but less strong signal is found in tree-ring widths. Growth in the previous year is also important (Briffa et al„ 1990).

month or months on which the tree-ring records are dependent, and which the tree rings can therefore be expected to usefully reconstruct. For example, by this approach Jacoby and D'Arrigo (1989) found that the ring widths of white spruce (Picea glauca) at the northern treeline in North America are strongly related to mean annual temperature, whereas Briffa et al. (1988, 1990) showed that summer temperature (April/May to August/September) is the major control on ring widths and maximum latewood density in Scots pine (Pinus sylvestris) in northern Fennoscandia.

Once the climatic parameters that influence tree rings have been identified, tree-ring data can be used as predictors of these conditions. Various levels of complexity may be involved in the reconstructions (Table 10.1). The basic level uses simple linear regression in which variations in growth indices at a single site are related to a single climatic parameter, such as mean summer temperature or total summer precipitation. An example of this approach is the work of Jacoby and Ulan (1982), where the date of the first complete freezing of the Churchill River estuary on Hudson Bay was reconstructed from a single chronology located near Churchill, Manitoba. Similarly, Cleaveland and Duvick (1992) reconstructed July hydrological drought indices for Iowa from a single regional chronology which was an average

TABLE 10.1 Different Levels of Complexity in the Methods Used to Determine Relationship Between Tree Ring Parameter and Climate

Number of variables of

TABLE 10.1 Different Levels of Complexity in the Methods Used to Determine Relationship Between Tree Ring Parameter and Climate

Number of variables of

Level |
Tree growth |
Climate |
Main statistical procedures used |

I |
1 |
1 |
Simple linear regression analysis |

Ha |
n |
1 |
Multiple linear regression (MLR) |

IIb |
nP |
1 |
Principal components analysis (PCA) |

Illa |
nP |
nP |
Orthogonal spatial regression (PCA and MLR) |

Illb |
nP |
nP |
Canonical regression analysis (with PCA) |

1 = a temporal array of data. n = a spatial and temporal array of data.

nP = number of variables after discarding unwanted ones from PCA. From Bradley and Jones (1995).

1 = a temporal array of data. n = a spatial and temporal array of data.

nP = number of variables after discarding unwanted ones from PCA. From Bradley and Jones (1995).

of 17 site chronologies of the same species. More commonly, multivariate regression is used to define the relationship between the selected climate variables and a set of tree-ring chronologies within a geographical area where there is a common climate signal. The tree ring data may include both ring widths and density values. Equations that relate tree rings (the predictors) to climate (the predictand) are termed transfer functions with a basic form (assuming linear relationships) of:

Vt = aiXlt + a2X2t + a3X3t - + an,Xmt + b + et where y is the climate parameter of interest (for year t); xlt, . . ., xmt are tree-ring variables (e.g., from different sites) in year t; ap . . . , am are weights or regression coefficients assigned to each tree-ring variable, b is a constant, and e is the error or residual. In effect, the equation is simply an expansion of the linear equation, yt = axt + b + et, to incorporate a larger number of terms, each additional variable accounting for more of the variance in the climate data (Ferguson, 1977). Theoretically, it would be possible to construct an equation to predict the value of yt precisely. However, adding too many coefficients simply widens the confidence limits about the reconstruction estimates so that eventually the uncertainty become so large that the reconstruction is virtually worthless. What is needed is an equation that uses the minimum number of tree ring variables to account for the maximum amount of variance in the climate record. Commonly, the procedure of stepwise multiple regression is used to achieve this aim (Fritts, 1962, 1965). From a matrix of potentially influential predictor variables, the one that accounts for most of the climate variance is selected; next, the predictor that accounts for the largest proportion of the remaining climatic variance is identified and added to the equation, and so on in a stepwise manner. Tests of statistical significance, as each variable is selected, enable the procedure to be terminated when a further increase in the number of variables in the equation results in an insignificant increase in variance explana tion. In this way, only the most important variables are selected, objectively, from the large array of potential predictors. An example of this approach is the reconstruction of drought in Southern California by Meko et al. (1980).

A major problem with stepwise regression is that intercorrelations between the tree-ring predictors can lead to instability in the prediction equation. In statistical terms this is referred to as multicollinearity. To overcome this, a common procedure is to transform the predictor variables into their principal modes (or empirical orthogonal functions, EOFs) and use them as predictors in the regression procedure. Principal components analysis involves mathematical transformations of the original data set to produce a set of orthogonal (i.e., uncorrelated) eigenvectors that describe the main modes of variance in the multiple parameters making up the data set (Grimmer, 1963; Stidd, 1967; Daultrey, 1976). Each eigenvector is a variable that expresses part of the overall variance in the data set. Although there are as many eigenvectors as original variables, most of the original variance will be accounted for by only a few of the eigenvectors. The first eigenvector represents the primary mode of distribution of the data set and accounts for the largest percentage of its variance (Mitchell et al., 1966). Subsequent eigenvectors account for lesser and lesser amounts of the remaining variance (Fig. 10.15). Usually only the first few eigenvectors are considered, as they will have captured most of the total variance. The value or amplitude of each eigenvector varies from year to year, being highest in the year when that particular combination of conditions represented by the eigenvector is most apparent. Conversely, it will be lowest in the year when the inverse of this combination is most apparent in the data. Eigenvector amplitudes can then be used as orthogonal predictor variables in the regression procedure, generally accounting for a higher proportion of the dependent data variance with fewer variables than would be possible using the "raw" data themselves. A time series of eigenvector values (amplitudes) is referred to as a principal component (PC), the dominant eigenvector being PCI, the next most common pattern PC2, etc.

Apart from reducing the number of potential predictors, principal components analysis also simplifies multiple regression considerably. It is not necessary to use the stepwise procedure because the new potential predictors are all orthogonal. This approach was used in the reconstruction of July drought in New York's Hudson Valley by Cook and Jacoby (1979). They selected series of ring-width indices from six different sites, and calculated eigenvectors of their principal characteristics. These were then used as predictors in a multiple regression analysis with Palmer Drought Severity Indices (Palmer, 19 6 5)32 as the dependent variable. The resulting equation, based on climatic data for the period 1931-1970, was then used to reconstruct Palmer indices back to 1694 when the tree-ring records began (Fig. 10.16). This reconstruction showed that the drought of the early 1960s, which affected the entire northeastern United States, was the most severe the area has experienced in the last three centuries.

32 Palmer indices are measures of the relative intensity of precipitation abundance or deficit and take into account soil-moisture storage and évapotranspiration as well as prior precipitation history. Thus they provide, in one variable, an integrated measure of many complex climatic factors. They are scaled from +4 or more (extreme wetness) to -4 or less (extreme drought) and are widely used by agronomists in the United States as a guide to climatic conditions relevant to crop production.

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Year uir lir 1000 170D 1BOO 1900 Year FIGURE 10.15 The first five eigenvectors of tree-ring widths based on a network of 65 chronologies distributed across the western United States, northern Mexico, and southwestern Canada.These represent the major patterns of growth anomalies in the region. Eigenvector I accounts for 25% of the overall variance; each subsequent eigenvector accounts for progressively less.The lower diagram shows the relative amplitudes of these five eigenvectors since A.D. 1600 (i.e., the principal components, [PCs] l-5).These and other PCs were used in canonical regressions with gridded temperature, precipitation, and pressure data, first to calibrate the tree-ring data and then to reconstruct maps of each climatic parameter back to A.D. 1600 (Fritts, 1991). Simple univariate transfer functions express the relationship between one climatic variable and multiple tree-ring variables. A more complex step is to relate the variance in multiple growth records to that in a multiple array of climatic variables (e.g., summer temperature over a large geographical region) (Table 10.1). To do so, each matrix of data (representing variations in both time and space) is converted into its principal modes or eigenvectors; these are then related using canonical regression or orthogonal spatial regression techniques (Clark, 1975; Cook et al., 1994). These i- i extreme T WETNESS X moderate T extreme I DROUGHT t moderate i extreme T WETNESS X moderate |

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