1 The Annual Ring—The Keeper of Time in Dendrochronology

In most parts of the world, the predictable and consistent march of the seasons, as our tilted planet Earth orbits around our sun, is strong enough to induce an annually-rhythmic time interval when environmental conditions are conducive to radial plant growth. Similarly, an annual rhythmic season when conditions are not conductive to growth also occurs, and plants for most dendrochronological purposes undergo a dormant season. This annual alteration of growing and non-growing seasons – perhaps one of the most regular features of the planet Earth on evolutionary time-scales—are often associated with changes in the types, and characteristics, of new wood cells developed over the course of a year (Fig. 2.1). These changes are associated with observable, often both macroscopically and microscopically, delineations in annual rings (e.g., Schweingruber 2001). In conifers (gymnosperms), the tracheid cells produced towards the beginning of the growing season tend to grow to larger sizes, yet have thin cell walls. As the growing season progresses and comes to a close, conifer tracheid cells tend to be smaller, particularly in the radial direction, and have thicker cell walls. In hardwoods (angiosperms), the number of wood cells and the associated structures tends to be more complex and variable, yet annual ring boundaries are also able to be differentiated based upon the characteristics of vessels (often larger and preferentially distributed towards the first part of a ring), and more dense fibers at the end of the growing season. Apart from annual temperature cycles, the monsoon that influences over half of the world’s population can also influence the formation of rings due to the prominent hydroclimate seasonality (e.g., Brienen et al. 2016). Especially in tropical regions, elemental composition and/or both stable (especially oxygen) and non-stable (especially carbon, i.e.,14C) isotopes have been used to quantify and confirm annual growth increments (Poussart et al. 2004, 2006; Anchukaitis et al. 2008; Xu et al. 2014). It is these annual growth increments, and our ability to distinguish them and ultimately identify and assign them to an exact year, that is the basis for dendrochronology.

Fig. 2.1
figure 1

Examples of wood anatomical and tree-ring structures for a gymnosperm (Abies alba), b a ring porous angiosperm (Quercus pubescens) c a diffuse to semi-ring porous angiosperm (Juglans regia), and d a diffuse to semi-ring porous angiosperm (Malus domestica). In all cases the outermost rings are towards the top of the images. The xylem of gymnosperms is composed primarily of tracheid cells, with a simpler and more uniform structure amongst species. Angiosperm xylem is composed of a several different types of cells whose characteristics and distributions within annual rings are highly variable amongst species resulting in a diverse range of visual appearances. A very well-prepared surface is necessary to differentiate annual tree-rings. Images from www.woodanatomy.ch (Schoch et al. 2004)

1.1 Inter-Annual Variations in Tree-Rings and Tree-Ring Parameters

The annual rhythm of growing and dormant seasons is the driving factor for annual ring formation. Yet it is the variations in external environmental forcing, predominantly inter-annual variations in weather and climate, that result in different ring traits and properties from one year to the next. Characteristic sequences of wider and narrower rings reflect changes in environmental conditions, and notably also results in a common signal or a unique fingerprint of this environmental variation amongst trees growing in proximal regions and ecological settings (Fritts 1976; Cook and Kairiukstis 1990; Speer 2010). Favorable climatic conditions for tree growth typically results in wider rings. Examples of favorable climatic conditions includes ample moisture for trees growing in more arid ecological settings, or sufficiently high temperatures conductive to photosynthetic activity and cellular growth for high elevation and high latitude trees. Whereas increased ecophysiological stress is typically associated with less radial growth. Although weather/climate variations play a crucial role for assigning the precise year to every ring (see Sect. 2.2), it should be noted that environmental conditions can and do include anything and everything that influences the molecular/isotopic composition, structure, ecophysiological functioning, and growth of trees. Tree-ring width is generally the most easily observable and measurable tree-ring parameter, yet many other tree-ring properties and parameters including wood density, (quantitative) wood anatomical assessments, and of course, stable isotopes, are influenced, often in unique ways, by environmental conditions. The plethora of tree-ring parameters thus expands the range of inter-disciplinary investigations possible with dendrochronology (Fig. 2.2). The rest of this section reviews various (non-stable isotope) tree-ring parameters, and provides an extensive listing of example research applications that have relied upon each of these parameters.

Fig. 2.2
figure 2

Illustration of tree-ring and wood anatomical parameters from a sample of Douglas Fir (Pseudotsuga menziesii) collected near Tucson, Arizona. Upper panel shows 11 complete annual rings with variability in the total tree-ring width as well as the earlywood and latewood widths. Also shown is a wood density profile based upon anatomical measurements of cell wall thickness using the program ROXAS (von Arx and Carrer 2014). Common ring-width parameters include tree-ring width, earlywood width, and latewood width (top). Important intra-ring density parameters include maximum latewood density and minimum latewood densities, as well as an intra-annual density fluctuation. Average ring density or average earlywood density (not shown in this figure) are also used in research applications (e.g., Babst et al. 2014b). Quantitative wood anatomical parameters include measures of lumen area, lumen diameter and cell wall thickness of the tracheids (bottom left). The lumen diameter and double wall thickness are shown in the radial direction—these can of course be measured in the tangential direction as well

1.1.1 Tree-Ring Width

A predominance of dendrochronological investigations performed to date have utilized assessments of the total tree-ring width. The ring-width has been long regarded as the basic unit of annual radial tree growth. Tree-ring width measurements are typically performed under a stereomicroscope connected to a linear encoder stage. In cases where rings can be sufficiently resolved with flatbed scanners, digital cameras, or imaged from thin sections, various software packages (e.g., Rydval et al. 2014; von Arx and Carrer 2014; Shi et al. 2019) can facilitate ring detection and measurement. Investigations relying upon tree-ring width span across the full range of inter-disciplinary applications including archaeological, climatological, ecological, and geological sciences. Examples include: timber trade and resource utilization (Guiterman et al. 2016; Domínguez-Delmás 2020); reconstructions of temperature primarily in higher latitude and/or altitude environments where cooler conditions limit radial tree-growth (Jacoby and D’Arrigo 1989; Esper et al. 2002; Salzer et al. 2014); assessments of past insect activity / outbreaks (Swetnam and Lynch 1989; Speer et al. 2001; Esper et al. 2007); reconstructions of drought (Cook et al. 2004; Fang et al. 2010), precipitation (Büntgen et al. 2011; Griffin and Anchukaitis 2014), streamflow (Woodhouse et al. 2006), and snowpack (Pederson et al. 2011; Belmecheri et al. 2016) variability primarily from locations where moisture limitations, potentially exacerbated by high thermal stress, limits radial growth; quantifying soil erosion rates (Gärtner et al. 2001; Rubiales et al. 2008); dating debris flows and avalanches (Stoffel and Bollschweiler 2008) as well as the construction of ancient (Douglass 1929) and historical structures (Boswijk et al. 2016); quantifying phenotypic traits, variability, and associations with genetic lineage/provenance (Housset et al. 2018); assessments of large-scale ecological disturbances (Pederson et al. 2014), and the resilience of ecosystems to climatic stressors (Chamagne et al. 2017; Kannenberg et al. 2019); understanding and predicting tree mortality (Cailleret et al. 2017); quantifying the climate sensitivity and drivers of forest growth in the past (Babst et al. 2013; St. George and Ault 2014), and making projections about tree growth in the future (Williams et al. 2013; Klesse et al. 2020); reconstructing larger-scale pressure and oceanic and atmospheric circulation indices (Trouet et al. 2009; Villalba et al. 2012; Li et al. 2013); quantifying forest carbon stocks and fluxes (Babst et al. 2014b); and integrating tree-ring and remotely sensed metrics of forest productivity (Coulthard et al. 2017; Seftigen et al. 2018).

Notably, tree-ring width is often divided into the earlywood and latewood widths (Fig. 2.2), sub-annual ring structures that have been demonstrated to carry distinct environmental signals in both angiosperms (Kern et al. 2013) and gymnosperms (Griffin et al. 2011). Measurement and analysis of these sub-annual ring components has allowed skillful reconstructions of the winter precipitation and summer precipitation to be developed from the earlywood and latewood width, respectively (Griffin et al. 2013). The distinct information carried in the intra-annual latewood and earlywood parameters is indicative of the relevance of finer-scale tree-ring parameters.

1.1.2 Tree-Ring Density

Zooming into a further level of magnification, a second overarching category of measurement parameters involves intra-annual variations in wood density (Fig. 2.2). Over 40 years ago, it was shown that high-resolution intra-annual wood density based upon x-ray measurements of conifer trees offered significant new opportunities for dendrochronology (Parker and Henoch 1971; Schweingruber et al. 1979). In particular the maximum latewood density carries extremely strong common signals among trees and sites (Schweingruber et al. 1979, Briffa et al. 2002a, Frank and Esper 2005). The maximum latewood density parameter has subsequently taken on a prominent role within the field of dendroclimatology for its skill in reconstructing warm-season temperatures (e.g., Hughes et al. 1984; Briffa et al. 2002b; Fan et al. 2009; Esper et al. 2012; Stoffel et al 2015; Wilson et al. 2016; Anchukaitis et al 2017). Meaningful and systematic environmental and ecophysiological signals in the earlywood density, that are notably distinct from tree-ring width and maximum latewood density, have gained increased attention recently (Camarero et al. 2014; Björklund et al. 2017; Buckley et al. 2018; Seftigen et al. 2020). Moreover, inter-annual density fluctuations or so-called false rings are connected to the ecophysiological responses to high aridity during the growing season, as well as to tree age and growth rates (Babst et al. 2016; Battipaglia et al. 2016; De Micco et al. 2016; Zalloni et al 2016).

Yet, the specialized equipment, labor intensiveness, and relative complexity of x-ray densitometry measurements has dampened the prevalence of density parameters. Regular flat-bed scanners, particularly utilizing blue channel intensity, offer the possibility to measure an optical density (McCarroll et al. 2002). Although there are potential challenges related to potential discoloration, surface preparation, heartwood/sapwood boundaries, and calibration with surface images, the climatic signals in the high frequency domain are comparable to those derived from x-ray-based techniques (e.g., Björklund et al. 2014, 2019; Wilson et al. 2019). However, it should also be noted that recent investigations and reviews (Jacquin et al. 2017; Björklund et al. 2019) have emphasized the equipment specific and resolution dependence on absolute values of the minimum and maximum density. Also crucial is the resolution and ring-width dependence on long-term (or apparent age/size related) trends in density measurements. These factors have potential implications for the divergence phenomena (Björklund et al. 2019; see also Wilmking et al. 2020 and references therein) and clearly merit further research.

At the other end of the technological spectrum, and related to classical x-ray densitometry, are newer applications that employ advanced 3d x-ray tomography (van den Bulcke et al. 2019). Such systems offers labor, analytical, and calibration advantages (Björklund et al. 2019). Other types of x-ray based measurements include measurements of the angle of microfibril in the cell wall via x-ray diffractometry. These microfibril measurements carry strong environmental signatures not reflected in tree-ring width or intra-annual density measurements (Drew et al. 2012). Also, x-ray micro fluorescence allows elemental mapping (see Sect. within annual rings at high resolution (Pearson et al. 2009; Sánchez-Salguero et al. 2019).

1.1.3 Quantitative Wood Anatomy

Tree-ring width and density parameters are, in fact, relatively simple functions of the number, sizes, distribution, and characteristics (e.g., cell wall thickness) of the wood cells (Vaganov et al. 2006; Fonti et al. 2010). For example, measurements of total tree-ring width in a given tree core are very highly correlated with the corresponding numbers of cells per ring. Thus, the types of investigations performed using quantitative wood anatomical approaches parallels, and in many respects expands upon, those performed using the tree-ring width and density parameters. Quantitative wood anatomical investigations similarly include climate reconstructions (Panyushkina et al. 2003; Ziaco et al. 2016), assessments of the environmental impacts on tree growth (Carrer et al. 2016; Lange et al. 2020; Puchi et al. 2020), and notably the seasonal progression of environmental influence from the beginning to the end of the annual growth flush (Castagneri et al. 2017; Popkova et al. 2018). The development measurement and analysis techniques for quantitative wood anatomy is contributing to paradigm shifts in tree-ring research that in many respects complements the overall subject of this book on isotopes and tree-rings. Specifically, these high-resolution investigations are contributing to viewing and quantifying the wood cell, and not the annual ring, as the fundamental building block of radial tree growth.

Moreover, wood anatomical studies are yielding deeper mechanistic insights into the growth processes and environmental responses of trees. For example, variations in inter-annual wood density can be well described and modeled as a consequence of competing cell enlargement and cell-wall thickening processes (Cuny et al. 2014; Björklund et al. 2017). Moreover, wood anatomy offers insights into ecophysiological processes related to drought stress and the turgor pressure required to develop and enlarge new wood cells (Friend et al. 2019; Cabon et al. 2020; Peters et al. 2021). Combining tree-ring, wood anatomical, and stable isotope data with complementary measurements from dendrometers (Deslauriers et al. 2007; King et al. 2013a), xylogenesis observations (Belmecheri et al. 2018; Delpierre et al. 2019), sapflow and sapwood area (Peters et al. 2019), offers more holistic assessments of ecosystem functioning, and opportunities to support terrestrial ecosystem and climate modeling approaches with in-situ and long-term data (Zuidema et al. 2018). Notably, it has been recently advocated to include the basic wood formation processes themselves into global vegetation models given the fundamental nature of these processes and the important role forest have on the global carbon balance (Friend et al. 2019).

Some quantitative wood anatomical measurements, for example of the larger vessels in some angiosperms (see Fig. 2.1), can be successfully performed using specifically prepared (e.g., with staining and/or filling vessels) core surfaces and flatbed scanning equipment (Fonti et al. 2009). However, more typically histological thin-sections and transmission microscopy (e.g., von Arx et al. 2016) complemented with specialized imaging and processing software (e.g., von Arx and Carrer 2014) are routinely applied. More specialized techniques such as confocal microscopy have been successfully employed (Ziaco et al. 2016), and there appears to be significant potential for high resolution 3D imaging methods with advancements in physical measurement technology as well as analytical and computation tools (van den Bulcke et al. 2019). Many wood anatomical investigations rely upon new analytical methods such as dividing annual rings into equal-width sectors (Castagneri et al. 2017), or into sequentially ordered and often standardized cell numbers (Popkova et al. 2018) via so-called tracheidograms (Vaganov et al. 2006; Peters et al. 2018). An emergent challenge appears to be to understand if and how the sub-annual timing associated with wood cells (e.g., Ziaco 2020) can be retrospectively inferred. If overcome, this will open up additional opportunities for multi-seasonal environmental reconstructions. At present, project-specific determinations on whether the added information from quantitative wood anatomical approaches outweighs the additional effort are warranted. However, with continued advancements in quantitative wood anatomy processing and analyses, it is likely that future dendrochronological studies will increasingly be based upon the cellular building blocks.

1.1.4 Isotopes and Wood Chemistry

For both brevity here, and completeness, the reader is referred to the rest of the book for a full overview on stable isotopes in tree-rings, as well as other sections in this chapter regarding specific implications to, and methods of, tree-ring research related to stable isotopes.

In terms of non-stable isotopes, dendrochronology has had (e.g., Leavitt and Bannister 2009), and continues to have (Pearson et al. 2018; Reimer et al. 2020) a crucial role in substantially improving the accuracy and precision of radiocarbon dating. Continued technological advancements (e.g., Synal et al. 2007) are permitting 14C measurements on long annual tree-ring sequences (Pearson et al. 2020), and continuing to reveal considerable inter-annual to decadal scale variability that is coherent at global-scales (Miyake et al. 2012; Jull et al. 2014; Büntgen et al. 2018). In addition to improving the radiocarbon calibration curve (Reimer et al. 2020), combined dendrochronological and radiocarbon investigations are yielding more, and often annually, precise dates that could not be achieved, at present, by either dendrochronology or radiocarbon studies alone (Oppenheimer et al. 2017; Kuitems et al. 2020; Pearson et al. 2020). Moreover, such coherent signatures in 14C amongst geographically distant sites, and without common climatic signals, represents a paradigm shift for tree-ring research (Dee and Pope 2016). In essence, this begins to create opportunities for “crossdating” at global scales (see Sect. 2.2 below), and amongst different proxy records (Sigl et al. 2015).

Although wood is primarily composed of carbon, oxygen, and hydrogen, an expansive range of trace elements are incorporated into the annual rings of trees. Subsequent elemental analysis may reflect influences from natural processes such as relatively recent (Sheppard et al. 2008) and ancient (Pearson et al. 2020) volcanic eruptions, be linked to anthropogenic influences such as heavy metals associated with industry (Muñoz et al. 2019), or depend upon year-to-year environmental variations and offer complementary information to the tree-ring width and density parameters (Sánchez-Salguero et al. 2019). Moreover, such measurements can serve as indicators of the annual cycle and help to define annual rings in tropical ecosystems (Poussart et al. 2006), and also provide indications of long-term trends that may be related to tree-age (Scharnweber et al. 2016),or combined climatic and anthropogenic variability (Panyushkina et al. 2016). At present it appears that the more specialized equipment, methods, and time consuming approaches contribute to rather modest sample replications for studies investigating trace elements in tree-rings. This seemingly makes combining such approaches with other independent data the most robust research pathway. Yet, it appears likely that with ever increased technological capacities, further methodological advancements, and consequently more highly replicated datasets a better understanding of the bioavailability and translocation/immobility will be achieved. This knowledge will presumably open up additional low risk and high reward research avenues.

1.1.5 Episodic Ring Features

Superimposed upon the normal progression of tree-ring formation (& analysis), episodic events can impact and leave distinct fingerprints within the annual rings. Such features can in turn be used to understand the nature of these, generally more sporadic, phenomena back in time. Inter-annual density fluctuations have already been discussed as evidence for mid-growing season drought that occurs almost every year in some ecological and climatic settings and very infrequently in others (Zalloni et al. 2016). Other types of episodic markers in tree-rings include so-called frost-rings, whereby freezing temperatures and the formation of ice crystals in the intercellular spaces damage developing wood cells (Barbosa et al. 2019 and references therein). Major cold snaps during the growing season that result in frost rings are often connected to major circulation and radiative forcing anomalies from large volcanic eruptions that eject aerosols into the stratosphere (LaMarche and Hirschboeck 1984; D’Arrigo et al. 2001; Salzer and Hughes 2007; Sigl et al. 2015). Tree-rings or especially latewood with anomalously light color (Tardif et al. 2011), or so-called “blue rings” (the term is derived from a common wood anatomical staining that presents incomplete lignification in a blue hue; Piermattei et al. 2015), similarly appear to be an indicator for especially cold conditions late in the growing season. Low-to-medium severity fires can create so-called fire scars that are evidence of localized cambial cell mortality and the subsequent healing over of these wounds. Individual trees often record many dozens of individual fires over their lifespans with composite records successfully used to reconstruct fire history, regimes and return intervals over the past millennia (Swetnam et al. 2009; Taylor et al. 2016). Similarly, traumatic resin ducts and/or compression and tension wood, can be indicative for localized injury or disturbances such as caused by past geomorphic/geological events (Jacoby et al. 1997; Stoffel and Bollschweiler 2008). Moreover, sudden transitions in tree roots, including wood anatomical properties, can be used to reconstruct erosion rates (Gärtner et al. 2001).

2 Crossdating

Annual ring formation, together with the characteristic coherent year-to-year variability in measurements of tree-ring sequences, gives rise to crossdating. Crossdating is easily the most fundamental concept and important method of dendrochronology, and can be broadly defined as the process by which the exact year is assigned to each and every ring (Stokes and Smiley 1968; Fritts 1976; Black et al. 2016). It is necessary to crossdate to ensure that all tree-rings, and all subsequent analyses, are properly aligned in time. Most typically crossdating results in the assignment of the exact calendar year to each and every ring. Exceptions to calendric dating exist with more ancient sites where a continuous tree-ring record to modern times has not yet been developed, or is not possible (Roig et al. 2001). While annual ring formation is determined by annually-cyclic growing and dormant seasons, crossdating is rooted upon intra-annual environmental variations, and especially years where strong growth reactions occur in response to more extreme environmental conditions.

Crossdating is necessary to: (1) distinguish inter-annual density fluctuations from annual ring termination (2) identify “locally absent” or so-called “missing rings” resulting from the lack of radial growth over the entire circumference and along the entire stem (and thus not present on the tree-ring sample/radius under investigation), (3) to identify extremely narrow rings e.g., those are only 1–2 cells wide that may have been initially overlooked, (4) to catch any possibilities for error in the assignment or measurement of rings, and (5) in the case of tree-ring materials spanning multiple generations (e.g., from historical or archaeological structures, dead relict wood preserved on the land surface, buried or submerged wood) to align these overlapping segments exactly in time. Only tree-ring sequences with a high degree of certainty in crossdating should be used for dendrochronological applications.

A variety of methods including skeleton plotting, graphical analyses, and statistical assessments of measurement series can be used to perform and assess the crossdating (Stokes and Smiley 1968; Fritts 1976; Holmes 1983; Cook and Kairiukstis 1990; Bunn 2010; Speer 2010). In practice, the development of tree-ring records typically proceeds in a hierarchical order—first working to ensure correct alignment among multiple cores from a given tree, and then amongst trees within a given site. The program COFECHA (Holmes 1983), as well as similar functionality within dplR package within the R programming environment (Bunn 2008; Bunn et al. 2020), are helpful to verify, or alternatively point to potential errors in, the crossdating for a given sample collection. Such programs allow rapid assessment of synchrony amongst tree-ring series in the hypothesized, as well as temporally lagged positions. The common signal, at least for tree-ring width and maximum latewood density, is typically most evident in stressful years that are associated with narrow rings (or low maximum latewood density) relative to the local mean of approximately one decade to either side of the ring in question. This has given rise to the practice to high-pass filter (i.e., remove multi-decadal to secular variability by detrending), remove autocorrelation, and logarithmically transform tree-ring measurement series prior to crossdating (Holmes 1983; Wigley et al. 1987).

Important considerations in confidently crossdating wood are the length (i.e., number of tree-rings) of records being examined, and the strength of the common signal. Accordingly, measures of synchrony such as the correlation coefficient, combined with information about the series length, can be used to statistically assess the confidence in a crossdated position (Wigley et al. 1987; Loader et al. 2019). The reader is specifically referred to Loader et al. 2019 for both an excellent discussion on the statistical tests for such assessments, and also the importance of tree-ring isotopes in crossdating (see also below). Any potential or hypothesized dating errors can often be conclusively identified by careful reexamination of the tree-ring samples. Pristine surface preparation is of critical importance. Proper use of successively finer grit sanding and polishing papers or alternatively razor blades / microtomes should allow the individual wood cells to be clearly observed under a microscope (Speer 2010).

Knowledge about the sample context, as well as auxiliary information such as the occurrence and temporal alignment of episodic features such as the presence of intra-annual density variations, frost rings, wide latewood, or traumatic resin canals can offer further confidence to the crossdated position determined from e.g., tree-ring width. Notably, continuous measurement series of wood density or stable isotopes, owing to their high-interseries correlation, can add greatly to the confidence in the assigned dates based upon ring-width (Frank and Esper 2005; Roden 2008; Hartl-Meier et al. 2014; Loader et al. 2019), and in fact open up opportunities to assign the correct dates to samples that could not be crossdated using tree-ring width along (Loader et al. 2019). Once site level crossdating has been achieved, additional independent tests of the crossdating can be garnered by comparison with other regional tree-ring chronologies, or even inferences based upon the expected patterns of tree growth from instrumental records or models of tree growth.

When working with particularly challenging sets of tree-ring samples such as from tropical regions, or on more ancient samples, radiocarbon dating can be performed to either refute (Herrera-Ramirez et al. 2017) or help verify annual ring formation and dating (Andreu-Hayles et al. 2015; Poussart et al. 2006). This represents an iterative advancement in the partnership between dendrochronology and 14C dating in which tree ring records, and increasingly annually-resolved measurements (Pearson et al. 2018), have long-been crucial to calibrate 14C dating. The potential to use multi-parameter approaches, including the high inter-series correlation that is becoming increasing evident in 18O measurements (Roden 2008; Hartl-Meier et al. 2014; Klesse et al. 2018c; Loader et al. 2019), appear to be a further promising direction for dendrochronological advancement.

3 Sampling and Site Selection

With the incredible multi-disciplinary range of dendrochronology (see Sect. 2.1.1 above), it is perhaps expected that the protocols and sampling designs to collect tree-ring data exhibit a wide range of variability amongst, and even within, sub-disciplines such as dendroclimatology, dendroecology, and dendroarchaeology (Fritts and Swetnam 1989; Cook and Kairiukstis 1990; Nehrbass-Ahles et al. 2014; Speer 2010). Ignoring the vast range in individual protocols, a typical sampling strategy would, for a location of interest, involve the collection of samples from e.g., two dozen trees. Such a collection of tree-ring samples is usually obtained from a geographically constrained area (e.g., within a few hundred meters), and where the trees are hypothesized to be in the same ecological setting, influenced by the same environmental factors, and usually of the same species. Such collections are typically crossdated and analyzed together, and referred to as a tree-ring site.

Increment borers are used to extract cores usually approximately 5 mm in diameter. For isotope measurements sometimes larger diameter (e.g., 1 cm) cores are collected to ensure enough sample material for subsequent analysis. Depending upon the research objectives and setting, cross-sections may be obtained and can be preferable especially in situations where samples are taken from trees that are no longer living. Even when the investigation may only involve isotope measurements for the most recent years, it is highly preferable for the collected tree core to span the full bark to pith radius. A full core allows estimates of tree age, and offers the greatest number of rings to facilitate crossdating. Collecting multiple cores per tree facilitates crossdating as well. Two cores per tree is routine. It is recommended to take even more cores per tree when investigating genera (e.g., Juniperus spp.) or locations (tropics) that are known to exhibit high within tree variability in the wood anatomical structures and/or ring characteristics. Specific investigations have not found a link between tree coring and increased mortality within multi-decadal time frames (van Mantgem and Stephenson 2004; Wunder et al. 2011).

Generally samples are collected at, or near, 1.37 m above the base of the tree. This corresponds to a routinely applied silvicultural measurement—the tree diameter at breast height (DBH)—and notably allows tree-ring width measurements to be more readily compared, and integrated, with periodic surveys of tree size (Evans et al. 2017). Moreover, collecting cores at this height: (i) mitigates growth irregularities /buttressing pronounced near the root collar, (ii) allows for the collection of most tree rings (rings produced when the tree was shorter than coring height will be missed), and (iii) is reasonably convenient for the physical collection of the sample (Cook and Kairiukstis 1990; Speer 2010). It is recommended to collect as many auxiliary data when sampling (e.g., tree diameter, height, location, canopy coverage), in order to assist with analysis and interpretation (see below on age/size/height trends), as well as integration into databases and subsequent multi-investigator compilations (Babst et al. 2014a; Brewer and Guiterman 2016).

Recent overviews of the International Tree-Ring Data Bank (ITRDB; Babst et al. 2017; Klesse et al. 2018b; Zhao et al. 2019; Pearl et al. 2020)—a compilation of thousands of tree-ring sites and datasets that have been graciously provided by hundreds of data contributors over the years in spirit of open exchange of scientific data—demonstrate that this collection is not representative of the global forested areas. Data in the ITRDB favor (i) gymnosperms over angiosperms, (ii) higher latitudes over tropical latitudes, (iii) older over younger trees, and (iv) are dominated by the tree-ring width parameter. In terms of tree-ring stable isotope measurements, at present only very few datasets have been submitted to the ITRDB. This presumably reflects the still expanding status of measurement efforts, the greater effort and expense in tree-ring isotope data, and far “younger” efforts to systematically share tree-ring isotope data (Csank 2009).

Sampling of tree-ring sites has been regularly coupled to the specific research question and objective. Recognizing the multiple influences on tree growth (see Sect. 2.4.1), there is a practice to select trees that are minimally influenced by environmental factors not under primary consideration (Fritts and Swetnam 1989; Cook and Kairiukstis 1990; Speer 2010). For example, when sampling trees for climate reconstruction, individual trees or forest stands whose growth is significantly influenced by factors such as geomorphic activity or obvious disturbance from insect outbreaks would tend to be avoided. For climate reconstruction, this would typically also involve the preferential sampling of trees where growth is most limited by the environmental parameter of interest. And for investigations of fire scars, sampling would focus on trees showing evidence of fire scars. Some, but not all, of the patterns of data collection in the International Tree-Ring Databank are driven by long-standing evidence that the common signal, crossdating, and also climatic information in tree ring sites, are strongest for trees growing near the distributional limits (Fritts 1976). This has been referred to as “The Principle of Tree and Site Selection”. So as described above, sampling for temperature reconstruction tends to take place near the upper or latitudinal treeline, whereas precipitation reconstructions tend to be developed from trees growing near the dry /lower treeline locations. This principle or paradigm of tree and site selection is abundantly supported by investigations utilizing tree-ring width, and amply supported by maximum latewood density. This paradigm, however, seems to be cast in doubt by tree-ring isotopes. There are an increasing number of studies that, in comparison to tree-ring width, show extremely strong common signals amongst proximal tree and sites for stable carbon and especially oxygen isotopes (e.g., Hartl-Meier et al. 2014; Klesse et al. 2018c) questioning the application of this principle to tree-ring sampling without consideration of the tree-ring parameters under investigation. It is plausible that a site selected to maximize the common signal in tree-ring width might not maximize the common signal for oxygen isotopes, and vice-versa. More thought, and empirical assessments, likely are warranted to revisit this topic with stable isotope data. It seems, however, that this should no longer be considered a universal principle for dendrochronology that aims to be inclusive for all tree-ring parameters.

4 Deconstructing Variability in Tree-Ring Data

4.1 The Linear Aggregate Model

A series of tree-ring measurements can be viewed as a composite of multiple underlying signals—namely all of the factors that influence the tree-ring parameters—whose individual contributions “sum-up” in the observed time-series (Fig. 2.3). The well-known linear aggregate model of tree growth (Cook 1987) formulates this perspective as an elegant conceptual framework that describes an observed tree-ring time series as the sum of the following five factors:

1) Biological age/size trend. Many measurements of tree growth change with the age and size of the tree. Some effects, for example, in tree-ring width measurements are geometric related to distributing a given amount of biomass around an ever-increasing circumference (Biondi & Qeadan 2008). Other age/size related trends may be due to the changing “social status” of trees within the stand and be related to relative access to light (Cook 1985; Klesse et al. 2018c), or perhaps roots to increasing soil depth with time. Many dendrochronology applications attempt to understand, and remove, the influence of age/size related trends in a process called “detrending” (see below).

2) Climate Signals. Inter-annual variability in climatic conditions such as temperature, precipitation, soil moisture, and vapor pressure deficit is at the heart of dendrochronology. It is the unique fingerprints of these variations in tree-ring parameters that enable crossdating. While crossdating usually focuses on the year-to-year variability, it should be noted that longer term (e.g., multi-decadal to centennial-scale and beyond) is also imprinted in tree-ring data. Such long-term variability is, however, more challenging to partition from age/size trends as we will see below. The climate signals tend to be common amongst trees at a site, as well as sites within broader regions (on the scale of hundreds of kilometers) provided similarities in the underlying ecological and climatic conditions (Fritts 1976; Schweingruber et al. 1979).

3) Endogenous Disturbance. Periodic disturbances to some trees at a given site due to tree or stand internal factors are also observed in tree-ring data. A classic example of an endogenous disturbance is related to gap phase dynamics in closed canopy forests. Here, growth pulses can be often observed in remaining trees following the removal (natural or anthropogenic) of nearby tree(s), especially if the tree(s) removed had a dominant canopy position and/or competed for moisture (Giuggiola et al. 2016). Provided the sampling covers a sufficiently large spatial scale relative to the size of the local disturbances, only some trees in a site will show responses to such disturbances. Endogenous disturbances may be the primary subject of dendroecological investigations (Pederson et al. 2014). Whereas for investigations where such disturbances are “noise”, their impact on tree-ring time series can be mitigated by a sufficiently large sample size collected from trees distributed over sufficiently large spatial areas, as well as statistical techniques (e.g., Principal Component Analyses or robust means) that can serve to extract signals common to all trees in a stand (Peters et al. 1981).

4) Exogenous Disturbance. Period disturbances affecting, most if not all trees in a stand, such as large-scale insect infestations, massive windthrow events, or fires are also evident in tree-ring data. Again, exogenous disturbances may be the primary topic of dendrochronological investigations. Both for applications aiming to retain or mitigate exogenous disturbance impacts, assessment of multiple sites across broader regions, different species (e.g., host versus non-host for reconstruction) as well as analytical techniques for disturbance detection and/or signal extraction are of value (Swetnam and Lynch 1989).

5) Error/Noise. The noise term of this model can be thought to include any variability not accounted for by the first four factors and is presumed to be uncorrelated amongst trees in a stand. Yet, clearly given the conceptual nature of the linear aggregate model, the “error” term in the originally conceived model may inherently including meaningful and scientifically understandable and interesting components such as genetic variation (King et al. 2013b; Housset et al. 2018). Mixed effects models build on the basic principles of the linear aggregate model. Yet importantly also allow for interactions and a non-independence of the factors (Klesse et al. 2020) with examples including a differential effect of climate on younger versus older trees, or compounded effects of drought induced stress and disturbance from insects on tree-ring width.

Fig. 2.3
figure 3

Illustration of the linear aggregate model. a Schematic example using fictitious data showing how an observed time-series can be decomposed as the sum of the age/size trend, climate induced variation, endogenous and exogenous disturbances, and some error term. The interannual variance of the climate term was decreased with increasing tree-age to represent the often observed decreased inter-annual variation in ring-widths as trees grow and distribute biomass around an ever increasing circumference. Note that the long-term trend in the climate term would be difficult to empirically partition from the age/size trend in a single observational time-series. b Examples of three time-series of tree-ring measurements scaled to have the same vertical range and aligned by innermost measurement which is assumed to represent a biological/cambial age (years since pith) of 1. The series are a Pinus ponderosa from New Mexico, USA (pink; Dean and Robinson 1986), Quercus douglasii from California, USA (orange; Stahle et al. 2012, 2013); USA, and Pinus sylvestris from higher latitudes in Sweden (green; Esper et al. 2012). The stronger multi-decadal variability evident in the high latitude site from Sweden is consistent with the temperature variations in this region. The Pinus ponderosa is a classic example for a negative exponential age/size trend as well as heteroscedasticity

4.2 Detrending and Standardization

While each of the terms in the linear aggregate model could receive its own chapter, and the reader is also referred to Cook (1987), a few additional paragraphs will be spent on the process of detrending and standardization. This is due to the importance of differentiating long-term age/size related trends from long-term environmental and ecological signals of interest for many dendrochronological investigations, and because active research focus is warranted as there does not appear to be a universal or best solution to detrend tree-ring data. However, it should be noted from the onset that potential advantages of stable isotope and some quantitative wood anatomical parameters are more moderate age/size related trends (i.e., noise for many applications) in comparison to tree-ring width. This topic is addressed in Sect. 2.4.3.

When aggregating multiple trees into a tree-ring chronology, it is often scientifically favorable to eliminate (to the maximum extent possible) the variation due to the age/size of a tree AND simultaneously retain all (to the maximum extent possible) variation from other factors of interest, say, climate. Moreover, differences in the absolute values of tree-ring measurements, say from a faster versus slower growing tree, or from one with lower versus higher 13C values, when developing a tree-ring chronology, also must be considered. Tree-ring detrending and standardization is the process by which time-series are handled, often sensu the linear aggregate model, to estimate and remove the age/size related trends and standardize tree-ring data into dimensionless indices (Fritts 1976; Jacoby and D’Arrigo 1989; Cook and Kairiukstis 1990; Cook and Krusic 2005; Melvin and Briffa 2008; Briffa et al. 2013).

There are two major categories of techniques to perform tree ring detrending, namely, those in which detrending curve is based upon (i) the individual tree-ring time series, or alternatively, (ii) a sample/population estimate of the age/size function. Most, if not all, of the earlier studies in dendrochronology relied on estimating the age/size trend on an individual, series-by-series, basis using deterministic (e.g., linear regression or negative exponential) curves that can be expressed as a simple equation, or alternatively, stochastic methods (e.g., moving averages, cubic smoothing splines, etc.) that are adapted to the data. Removal of the age/size trend is then typically achieved by dividing the observed (i.e., the tree-ring measurements) by the estimated (i.e., either the deterministically or stochastically fit curve) resulting in dimensionless tree-ring indices. Individual detrending may be performed with “conservative” methods (e.g., linear regression or negative exponential curves) that attempt to retain longer-term climate information (e.g., Jacoby and D’Arrigo 1989) or alternatively flexible curves that primary retain inter-annual to decadal-scale variability as is desirable for crossdating (e.g., Holmes 1983). In addition to removing age/size related trends, detrending by division (i.e., ratios) has the benefit of mitigating the commonly occurring heteroscedastic relationships in tree-ring width data namely the often observed relationship between the local variance and mean (or spread versus level). Dividing the tree-ring measurements by the growth curve results in a local normalization of variance structure along the tree-ring series (see Fig. 2.4). For maximum latewood density data (e.g., Helama et al. 2008), and likely also stable isotope data (Büntgen et al. 2020), where such heteroscedasticity is clearly weaker, detrending can be performed by subtracting the modeled aged trend from the tree-ring measurements (e.g., Cook and Peters 1997). Importantly, this detrending, also standardizes tree-ring series to eliminate influences from difference in absolute growth rates (or isotope discrimination) in a tree-ring chronology as all tree-ring data are now in dimensionless indices with a mean of approximately unity.

Fig. 2.4
figure 4

Schematic illustration of potential challenges in identification of growth trends in tree-ring data with e.g., implications for assessment of CO2 fertilization effects. The top panel shows a fictitious example of tree-ring width data (blue) with an age/size related trend. Examples of methods applied in the literature to assess growth changes including (i) no detrending (note this method appears useful for the last millennia of multi-millennial tree-ring series where strong conclusions are able to be drawn if an increase in tree-ring width is observed; Salzer et al. 2009), (ii) flexible detrending, (iii) conservative detrending (see also Peters et al. 2015) also with some degree of end-fitting artifacts (see Cook and Peters 1997) are shown. The manner in which these illustrative tree-ring data have been handled impacts potential conclusions on growth trends. This example aims to illustrate the relevance of careful assessments of potential age/size related trends in analysis, interpretation, and dissemination. Note, dividing the tree-ring data with fit curves results in a homoscedastic tree-ring indices in the lower two panels

Three well-recognized pitfalls exist with individual detrending. Firstly, it is not possible to confidently differentiate age/size related trends from other signals of interest such as climate (but see “signal free” detrending discussion below). Secondly, variability on time-scales longer than the individual tree-ring measurement series cannot be retained—the so-called “Segment Length Curse” (Cook et al. 1995)—which is especially problematic for investigations that build multi-centennial or millennial length records from shorter, overlapping, crossdated series. Lastly, biases in the tree-ring indices might be created when the estimated curve approaches values close to zero (Cook et al. 1997). The recognition of these pitfalls, and especially the first two, challenges has driven continued innovations in detrending methods—particularly when long-term (e.g., centennial-scale and beyond) information is sought (e.g., Briffa et al. 2013).

The second category of established methods to remove the age/size related trends, and one in a manner that can preserve low-frequency variability and break the “Segment Length Curse”, is the so-called “Regional Curve Standardization” method (RCS; Briffa et al. 1992; Cook et al. 1995; Esper et al. 2003; D’Arrigo et al. 2006; Helama et al. 2017). The RCS method allows retention of low-frequency signals by detrending all series with the same regional growth curve. Detrended series are thus not constrained to a mean of unity. Rather, they can “float above” or “sink below” the regional mean to capture periods of above or below average growth on time-scales longer than individual trees’ lifespans. In RCS, a population estimate, the so-called regional curve, for the age-related trend is estimated based upon all tree-ring data at a site. Operationally, the regional curve is estimated by aligning all tree-ring series by their cambial age (i.e., first ring after the pith is a cambial age of 1) and averaging the age-aligned data together. In cases where the tree-core (or cross-section) does not include the pith, the number of years to the pith can (and should) be estimated (see Pirie et al. 2015 for a review). Methods for RCS employed in the ARSTAN (Cook and Krusic 2005) and dplR (Bunn 2008; Bunn et al. 2020) frameworks both allow the inclusion of pith-offset information. The detrended tree-ring data are then derived by removing (usually by ratios for tree-ring width data) the regional curve from all measured tree-ring series.

RCS application ideally is performed on tree-ring records (1) with a diverse range of start (pith) and end dates (outermost ring), (2) that are very highly replicated with many individual samples, and (3) from a homogeneous well defined ecological setting where all trees are expected to have similar growth rates and ontogenetic trends (e.g., Esper et al. 2012; Briffa et al. 2013). The first point is important because this helps mitigate the impact of temporally aligned events (such as climatic fluctuations or ecological disturbances) in the calendrically dated tree-ring data from impacting the shape of the regional curve. Secondly, composite datasets comprised of overlapping crossdated sequences for the past millennia from a single, well constrained ecological setting appear ideal for the application of RCS as a single regional curve could theoretically be used (Briffa et al. 2013). Well-replicated datasets are especially critical as the tree-ring indices are not constrained to a mean of one. On one hand, the differences in growth rates that are standardized in individual detrending add noise and greater uncertainty about the mean (e.g., D’Arrigo et al. 2006; Melvin and Briffa 2014b). But on the other hand, the subtle differences in mean growth rates of cohorts of trees that lived at different times is precisely the low-frequency signal that can be retained with RCS (D’Arrigo et al. 2006). And lastly, a homogenous ecological setting is crucial to ensure both uniformity in the underlying climate signal as well as growth rates. This criterion is difficult to ensure, but assessments of the growth rates of different cohorts over time might suggest homogeneity. In cases where there is reason to doubt the homogeneity of a dataset, it is feasible to develop and apply regional curves for distinct dataset subsets, e.g., the modern and ancient portions of a tree-ring chronology. Detrending via multiple RCS cohorts should be performed in situations where sampling differences, site difference, or anthropogenic disturbances (e.g., forest management regimes, responses to elevated CO2 or nitrogen concentrations) may significantly compromise the homogeneity of the dataset over time. Such approaches have the advantage of reducing potential biases in the final chronology, but at the same time make it more difficult to retain long-term variability of interest (Briffa et al. 2013).

A third approach for detrending that continues to gain traction in tree-ring research is the so-called “Signal Free” method (Melvin and Briffa 2008; Melvin and Briffa 2014a, b). This approach is based upon the recognition that the curves used to detrend tree-ring data, particularly for individual detrending, do not (readily) distinguish between the age/size trends and long-term climatic variability (Melvin and Briffa 2008). Conceptually, the “signal free” approach is similar to the RCS. Yet, instead of developing a population estimate of the regional curve, a population estimate of the common non-age/size related variability (i.e., generally the climate signal) has the central role. Specifically, the signal free method iteratively removes a population estimate of the tree-ring chronology from the tree-ring measurements prior to the next detrending iteration. Multiple iterations are performed until (in most cases) a stable estimate for the tree-ring data and chronology is achieved. The “signal free” method can be applied at the individual series level, or alternatively can be employed to derive a signal-free estimate of the regional curve in RCS. Investigations that have employed the signal-free detrending have shown a superior retention of lower-frequency climate related signals in the final tree-ring chronology (e.g., Fang et al. 2012; Melvin and Briffa 2014a; Wilson et al. 2019).

Awareness of the potential age/size related trends is crucial for all tree-ring and subsequent detrending/standardization approaches. Moreover, awareness of how such long-term trends, and/or methodological approaches applied to remove them (Fig. 2.4), potentially impacts scientific investigations is also required, as is communicating such information in publications. Continued research on both methodological detrending/standardization methods, and also emphasizing the importance of highly replicated tree-ring records, are both necessary.

4.3 Long-Term Trends in Tree-Ring Data

Tree-ring data are of great value for the centennial to multi-millennial scale perspectives on changing ecosystems, climate, and societies, including rare, extreme, and long-term variations. A recurring theme in the dendrochronology literature is the faithful decomposition—sensu the linear aggregate model—of the variable of interest while mitigating noise from other factors. As described in Sect. 2.4.3 much of the focus in dendrochronology literature has been on removing age/size related trends, while retaining all long-term climate related trends. This endeavor is particularly challenging due to i) multiple environmental and biological factors and processes that could result in long-term trends in tree-ring data, ii) reduced degrees of freedom to statistically assess and verify lower-frequency changes (e.g., Briffa et al. 2002a; Melvin and Briffa 2014b) and iii) limited opportunities to make direct comparisons with independent long-term records of the same exact phenomena (Emile-Geay et al. 2017) recognizing further differences in seasonality, resolution, and geographic proximity. As dendrochronology continually expands by covering longer periods of time, over more regions, into newer disciplines such as terrestrial carbon cycling and ecophysiology, and with greater reliance on parameters from wood anatomy and stable isotopes, new challenges and opportunities are arising. There are a perhaps a couple of points of note related to these developments, and with a focus on stable isotopes.

Firstly, there is not a consensus on the existence or absence of age/size related trends in stable isotopes (e.g., Klesse et al. 2018c and references therein; Büntgen et al. 2020 and references therein; McCarroll et al. 2020 and references therein), and thus also not for the potential mechanisms of such age/size trends nor how they may differ for carbon and oxygen isotopes. Given the evidence and disparities in the literature, it seems that the following working conclusions can be drawn:

  1. (1)

    Investigators should consider the possible existence of age/size related trends in their data, how such potential trends are best quantified, and attempt to identify if and how such methodological approaches related to removing (or not) potential age/size trends, may impact the subsequent analysis and conclusions.

  2. (2)

    Age/size/height related noise in most tree-ring stable isotope data appear to be a smaller fraction of the variance in comparison to most tree-ring width data. On one hand, this suggests less than perfect removal, if needed, has less severe consequences on the study. Yet this also suggests, it is more difficult to identify, consider and remove age/size/height related trends.

  3. (3)

    Even in the absence of age/size/height trends in tree-ring stable isotope data for a given location and species, the substantial offset in absolute values both within a single tree and amongst different trees at a site (e.g., Leavitt and Long 1986; Klesse et al. 2018c; Esper et al. 2020), indicate standardization and/or thoughtful compositing is required to minimize signals that might be artifacts as individual time-series enter or exit from a tree-ring chronology (e,g, Hangartner et al. 2012; Melvin and Briffa 2014b)

Secondly, the use of 13C measurements as indicators for both direct climate effects on trees such as, temperature, precipitation and sunshine variation (e.g., Hafner et al. 2014), and to also infer changes in water use efficiency (e.g., Saurer et al 2004), appears to require more efforts to robustly separate CO2 and meteorological driven changes in carbon isotope discrimination. Existing approaches include constrained stochastic detrending (McCarroll et al. 2009) and the use of climate variability (Treydte et al. 2009; Frank et al. 2015, but see Lavergne et al. 2019). Further development of well-replicated tree-ring 13C, 18O and tree-ring width / wood anatomy datasets and complemented by detailed ecological / forest biometric data (e.g., Klesse et al. 2018a), as well as integrated empirical-modelling approaches, as facilitated by this book, will offer needed insights and opportunities to disaggregate and understand the drivers of isotope discrimination and forest ecosystem functioning.

5 Chronology Development, Confidence, Sample Replication, Coherence, and Variance

5.1 Tree-Ring Chronologies

In comparison to other earth and environmental science related fields, the relative ease with which tree-ring samples are collected and measured has greatly facilitated well-developed statistical frameworks for signal and noise assessment in dendrochronology (e.g., Wigley et al. 1984; Cook and Kairiukstis 1990). Multiple cores per tree, multiple trees per site, and multiple sites per region, can be regarded as hierarchical view on drawing samples from a population, and using such samples to estimate population characteristics. Within dendrochronology it is typical, after crossdating, to take the multiple individual measurement time-series and average (or use another statistical measure of central tendency such as the robust mean, median or other percentiles) their values into a single time-series—a tree-ring chronology. A tree-ring chronology enjoys the characteristic that the common signal from the sampled population of individual trees is enhanced by averaging out “noise” variance specific to individual time-series.

The most basic type of tree-ring chronology consists of the average of all individual tree-ring time-series from a given site. This chronology (e.g., referred to in ARSTAN (Cook and Krusic 2005) as the “Raw” chronology) incorporates all of the underlying signals in the tree-ring data including possible age-size related trends, variance changes due to the heteroscedastic nature of tree-ring data, potentially artifacts from changes in sample replication (see below), and so on. Yet, many investigations derive chronologies after performing data processing and analytical steps (sensu the linear aggregate model) to remove such unwanted noise. Thus, typically the tree-ring data are detrended & standardized (see above) prior to averaging together into most tree-ring chronologies. It is broadly recognized there is no objective or perfect way to detrend the tree-ring data, and thus an emphasis on both understanding and communicating in publications how the detrending impacts the retained signals in the final chronology is necessary. When emphasis is on year-to-year variations, as for crossdating, more flexible detrending curves can be used to remove most variability on say time-scales longer than 30-years whilst retaining essentially all year-to-year variations. Similarly, when the preservation of long-term trends is important, as is the case for many global change related investigations, more conservative approaches to retain longer time-scale variability are required (e.g., Jacoby and D’Arrigo 1989; Briffa et al. 1992; Cook et al. 1995; Esper et al. 2002; Melvin and Briffa 2008, 2014a, 2014b; Helama et al. 2017). Serial autocorrelation, or the statistical non-independence of successive years of tree growth due to e.g., biological factors such as carbohydrate reserves (or depletion) and changes in needle/foliar length and capacity, is sometime also removed from tree-ring measurements in a step often referred to as “pre-whitening” via auto-regressive moving average modeling (Meko 1981). It is also standard practice to make a mean chronology with a bi-weight robust mean to mitigate the influence of statistical outliers (Cook 1985). Some recent work has also suggested that other estimates of central tendency and/or dispersion such as percentile chronologies may offer distinct advantages to mitigate noise that is not uniform to all trees in all years (Stine and Huybers 2017).

Within dendrochronology the program ARSTAN (Cook and Holmes 1986; Cook and Krusic 2005) is extremely well established for performing all of the steps related to chronology development and statistical assessments of chronology quality (see below). The dplR package (Bunn 2008; Bunn et al. 2020) in the R programming environment has replicated many of the steps in ARSTAN versions, and offers the advantage of ease of integration with other user-defined analytical or graphical procedures. Signal-free detrending can be performed using CRUST (Melvin and Briffa 2014a, 2014b) or ARSTAN variants. Many of the basic procedures can also be performed in user developed algorithms, including potential mixed-effect models which offer innovation potential for the discipline. However, such user-specific approaches may be less well understood by the broader scientific community, and would require elaborate development to meet robust best practices for both chronology development and assessment (see below).

5.2 Assessment of Chronology Confidence

It is desirable to understand how well the tree-ring data and chronology represents the site, or the theoretical population, that they purportedly sample. In this regard, two important quantities are: (i) an assessment of the correlation or common signal of the tree-ring measurements and (ii) the sample size. In the simplest case where one core is collected per tree, the average correlation coefficient, \(\overline{{\varvec{r}} }\), among all possible time-series pairs is a good estimate for the common signal. Understanding what components of the tree-ring data (e.g., age-related trends, autocorrelation, pre-whitened time-series) are included or not in correlation computations is however needed. Similarly, the number of trees (= the number of tree-ring time series) at any given time would be an assessment of the sample size. Yet, the common practice to collect multiple cores per tree somewhat complicates the assessment of the signal and noise (Fritts 1976; Wigley et al. 1984; Briffa and Jones 1990), and thus requires a bit more consideration. In this regard, the first term which may be defined is the effective average number of cores per tree,\({{\varvec{c}}}_{{\varvec{e}}{\varvec{f}}{\varvec{f}}}\), at any given time:


where \({{\varvec{c}}}_{{\varvec{i}}}\) is the number of cores from tree i, for all n trees at a site. \({{\varvec{c}}}_{{\varvec{e}}{\varvec{f}}{\varvec{f}}}\) typically will change over-time as e.g., tree cores that do not come close to the pith no longer contribute data in earlier years. More interesting is, however, the assessment of the correlation (usually either via Pearson’s r or Spearman rho). Notably, the correlation amongst all possible pairs of time-series includes those computed from within the same tree, and those computed among different trees. Typically, measurement series from a single tree will tend to be more highly correlated with each other than measurements series from different trees. Counter-intuitively higher within tree correlations are due to tree-specific noise that, if not accounted for, could lead to inflated confidence in a tree-ring chronology particularly with (i) greater numbers of cores per tree and ii) fewer number of trees per site. An accurate assessment of common signal thus requires computing separately: the average correlation \({(\overline{{\varvec{r}}} }_{{\varvec{t}}{\varvec{o}}{\varvec{t}}})\) amongst all pairs of time series (\({{\varvec{P}}}_{{\varvec{t}}{\varvec{o}}{\varvec{t}}})\); the average correlation (\({\overline{{\varvec{r}}} }_{{\varvec{w}}{\varvec{t}}}\)) for the \({{\varvec{P}}}_{{\varvec{w}}{\varvec{t}}}\) pairs of times series computed within the same tree; and the average correlation (\({\overline{{\varvec{r}}} }_{{\varvec{b}}{\varvec{t}}}\)) for the \({{\varvec{P}}}_{{\varvec{b}}{\varvec{t}}}\) pairs of times series computed between different trees. Typically \({\overline{{\varvec{r}}} }_{{\varvec{b}}{\varvec{t}}}\) is calculated as:


The difference between \({\overline{{\varvec{r}}} }_{{\varvec{w}}{\varvec{t}}}\) and \({\overline{{\varvec{r}}} }_{{\varvec{b}}{\varvec{t}}}\) is an assessment of the tree-specific noise. Finally, the effective average inter-series correlation, \({\overline{{\varvec{r}}} }_{{\varvec{e}}{\varvec{f}}{\varvec{f}}}\), can be computed as:

$${\overline{{\varvec{r}}} }_{{\varvec{e}}{\varvec{f}}{\varvec{f}}}=\frac{{\overline{{\varvec{r}}} }_{{\varvec{b}}{\varvec{t}}}}{{\overline{{\varvec{r}}} }_{{\varvec{w}}{\varvec{t}}}+\frac{1-{\overline{{\varvec{r}}} }_{{\varvec{w}}{\varvec{t}}}}{{{\varvec{c}}}_{{\varvec{e}}{\varvec{f}}{\varvec{f}}}}}$$

When \({{\varvec{c}}}_{{\varvec{e}}{\varvec{f}}{\varvec{f}}}\) = 1, \({\overline{{\varvec{r}}} }_{{\varvec{e}}{\varvec{f}}{\varvec{f}}}\) = \({\overline{{\varvec{r}}} }_{{\varvec{b}}{\varvec{t}}}\); and similarly as \({{\varvec{c}}}_{{\varvec{e}}{\varvec{f}}{\varvec{f}}}\) becomes very large, \({\overline{{\varvec{r}}} }_{{\varvec{e}}{\varvec{f}}{\varvec{f}}}\) approaches the \({\overline{r} }_{bt} to{\overline{ r} }_{wt}\) ratio. For anecdotally reasonable values of \({{\varvec{c}}}_{{\varvec{e}}{\varvec{f}}{\varvec{f}}}\) = 1.9, \({\overline{{\varvec{r}}} }_{{\varvec{w}}{\varvec{t}}}\) = 0.7, and \({\overline{{\varvec{r}}} }_{{\varvec{b}}{\varvec{t}}}\) = 0.5 the above equation yields an \({\overline{{\varvec{r}}} }_{{\varvec{e}}{\varvec{f}}{\varvec{f}}}\) of 0.583 reflecting the added value of the multiple cores per tree on top of the \({\overline{{\varvec{r}}} }_{{\varvec{b}}{\varvec{t}}}\) = 0.5. With the above terms at hand, it is now possible to estimate how well the particular sample of tree cores represents the theoretical population chronology from which it is drawn. Following Wigley et al. (1984) and Briffa and Jones (1990) the Expressed Population Signal (EPS) can be defined as:

$${\varvec{E}}{\varvec{P}}{\varvec{S}}\approx \frac{{\varvec{N}}{\overline{{\varvec{r}}} }_{{\varvec{e}}{\varvec{f}}{\varvec{f}}}}{(1+\left({\varvec{N}}-1\right){\overline{{\varvec{r}}} }_{{\varvec{e}}{\varvec{f}}{\varvec{f}}})}$$

where N is the number of trees (not cores) that are included in the dataset. EPS values approaching unity approach the theoretical population signal. These same two works also derived and discussed the related statistic, the Subsample Signal Strength (SSS). The SSS defines how well a subset of trees represents the full tree sample. The SSS is relevant as the number of trees varies typically with fewer and fewer trees further back in time. The SSS is defined as:

$${\varvec{S}}{\varvec{S}}{\varvec{S}}\approx \frac{{\varvec{n}}(1+\left({\varvec{N}}-1\right){\overline{{\varvec{r}}} }_{{\varvec{e}}{\varvec{f}}{\varvec{f}}})}{{\varvec{N}}(1+\left({\varvec{n}}-1\right){\overline{{\varvec{r}}} }_{{\varvec{e}}{\varvec{f}}{\varvec{f}}})}$$

where \({\varvec{n}}\) is the subset of trees in a chronology at any given point in time. The SSS can also be expressed and calculated as:

$${\varvec{S}}{\varvec{S}}{\varvec{S}}=\frac{{\varvec{E}}{\varvec{P}}{\varvec{S}}({\varvec{n}})}{{\varvec{E}}{\varvec{P}}{\varvec{S}}\boldsymbol{ }({\varvec{N}})}$$

Notably, the EPS and SSS statistics are a simple function of the two terms – sample size and common signal—that are fundamental to assessing chronology quality. With increases in sample size, a greater proportion of tree and core specific noise is averaged out of the mean chronology, resulting in better and better estimates of the population mean. Similarly, higher correlations between trees are indicative of a strong common signal and lower amounts of noise, with fewer trees generally required (but see below discussion on lower-frequency variation) to achieve a certain robustness. The connections between the theoretical approach of Wigley et al. (1984) and earlier more empirically-based analysis of variance approaches (Fritts 1976) to assess chronology signals, noise, and confidence, have been shown to be essentially identical if the data are normalized (Wigley et al. 1984; Briffa and Jones 1990). It remains outstanding, to the best of our knowledge, to demonstrate the potential equivalence or differences between these statistical frameworks and mixed effects modeling approaches that have gained traction in many environmental and ecological fields in recent years.

The Wigley et al. (1984) paper that introduced the EPS and SSS statistics provided illustrative examples of how such statistics can be used to assess chronology confidence using an illustrative value of SSS = 0.85. This example was subsequently expanded upon by Briffa and Jones (1990). Within the broader community there has been widespread use of this illustrative threshold 0.85 for EPS and SSS (see Buras 2017) as a criteria to demonstrate chronology robustness. While recognized as arbitrary threshold for the SSS (Wigley et al. 1984) and EPS (Briffa and Jones 1990), this value is also arguably an objective criteria that allows comparisons of chronology confidence among studies (Briffa and Jones 1990). Figure 2.5 illustrates how EPS and SSS change as a function of sample size and correlation. Some notable aspects are: (1) both EPS and SSS rapidly achieve relatively high values at low numbers of series, particularly for higher \({\overline{{\varvec{r}}} }_{{\varvec{e}}{\varvec{f}}{\varvec{f}}}\) values; (2) “returns” on additional samples diminish at higher samples sizes (but again see below discussion on lower-frequency variation); (3) the EPS is a more conservative estimate (i.e., more series are required to achieve a given threshold) of absolute chronology robustness; (4) the SSS and EPS are quite similar when the underlying tree-ring data are highly correlated.

Fig. 2.5
figure 5

This figure displays calculations of the Expressed Population Signal (EPS, dots) and Subsample Signal Strength (SSS, lines) statistics for two different levels (0.75 and 0.25; dark and light shades respectively) of effective interseries correlation from illustrative total sample sizes of 15 (purple tones) or 50 (green tones) trees

Perhaps less widely appreciated, is that these statistics, due to their correlation-based nature, do not assess chronology confidence (or error) that results from absolute differences in the scale or variance of the underlying time-series (Briffa and Jones 1990; Melvin and Briffa 2014b). Furthermore, these statistics are only moderately sensitive to low-frequency (in)coherence, particularly when computed in moving (e.g., 50-year) windows (Melvin and Briffa 2014b). For many highly correlating tree-ring parameters such as tree-ring width from semi-arid or high latitude/altitude sites (e.g., Breitenmoser et al. 2014; St. George and Ault 2014), maximum latewood density from high latitude/altitude sites (Briffa et al. 2002a, b; Björklund et al. 2017), and notably stable isotopes across a wide range of ecological and climatic environments (Hartl-Meier et al. 2014), achieving illustrative EPS and SSS benchmarks of e.g., 0.85 may require only 4–5 measurement time-series. Yet, more emphasis appears warranted to address and communicate these estimates of chronology confidence are for (primarily) high-frequency assessments. Similarly, more work (and more samples) are required for any dendrochronological investigations benchmarking longer-term variability. For example, Melvin and Briffa (2014b) describe an approach to estimate lower-frequency coherence by looking at variance ratios of the data before and after high-pass filtering. They suggest that for lower-frequency assessment several times more trees might be actually necessary than inferred from the EPS and SSS calculations. Anecdotally, trend discrepancies amongst investigations of tree-ring isotopes support the notion that many more series will be scientifically important. Thus, two major research agendas are: (1) to better quantify the skill and error especially towards lower-frequency domains, and (2) to develop ever larger tree-ring datasets to increase confidence (and more realistically assess noise) in lower-frequency domains. The former has been a challenge to the paleoclimatic community for decades (e.g., Esper et al. 2004), and becoming a clearly recognized need for assessments of stable isotopes in narrowing the wide range of water use efficiency responses to the anthropogenic increase in CO2 (e.g., Saurer et al. 2004). Yet, with continued advances in technology and processing efficiency, and decreases in expenses (e.g., Andreu-Hayles et al. 2019), it is foreseeable that we will soon have many more tree-ring isotope datasets with 10’s of trees at any given point in time (with less or no pooling across trees) and thus steady progress towards realizing the latter research agenda.

5.3 Variance Changes in Composite Time-Series

The same fundamental quantities, the sample size and coherence amongst series, also determine the variance, \({{\varvec{S}}}^{2}\), of the resulting dataset average—the tree-ring chronology (Wigley et al. 1984; Osborn et al. 1997; and see also Shiyatov et al. 1990). Following Wigley et al. (1984) and Osborn et al. (1997) this relationship can be expressed as:

$${{\varvec{S}}}^{2}=\overline{{{\varvec{s}} }^{2}}\left[\frac{1+({\varvec{n}}-1){\overline{{\varvec{r}}} }_{{\varvec{e}}{\varvec{f}}{\varvec{f}}}}{{\varvec{n}}}\right]$$

where \(\overline{{{\varvec{s}} }^{2}}\) is the mean variance of the individual tree-ring time-series, and \({\varvec{n}}\) is the number of trees (at any given time) and \({\overline{{\varvec{r}}} }_{{\varvec{e}}{\varvec{f}}{\varvec{f}}}\) is defined above. Notably, changes in sample replication and/or series coherence over time will cause the variance of the mean chronology to also vary. While series coherence may vary due to a number of reasons (including those that should be retained in the tree-ring chronology), any variance structure that depends upon sample size must be regarded as an artifact of the sampling and should thus be removed from the final tree-ring chronology (Osborn et al. 1997; Frank et al. 2007). As tree-ring time-series are generally significantly correlated with each other—a general outcome of the common signal and the basis for crossdating—the effective sample size,\({{\varvec{n}}}_{{\varvec{e}}{\varvec{f}}{\varvec{f}}}\), can be defined as:

$${{\varvec{n}}}_{{\varvec{e}}{\varvec{f}}{\varvec{f}}}=\frac{{\varvec{n}}}{1+({\varvec{n}}-1){\overline{{\varvec{r}}} }_{{\varvec{e}}{\varvec{f}}{\varvec{f}}}}$$

which is simply the reciprocal of the bracketed part of the above equation. In this way \({{\varvec{n}}}_{{\varvec{e}}{\varvec{f}}{\varvec{f}}}\) serves at the factor by which a mean chronology can be multiplied to yield a chronology corrected for variance artifacts resulting from changes in sample depth. And specifically, multiplication of the mean chronology (after centering about a mean of zero) by \(\sqrt{\overline{{\varvec{r}} }\boldsymbol{*}{{\varvec{n}}}_{{\varvec{e}}{\varvec{f}}{\varvec{f}}}}\) yields a mean chronology whose variance is adjusted and the scaling/units are preserved to the variance that would result from the theoretical population chronology (Osborn et al. 1997; Frank et al. 2007). While handling the variance artifacts from changes in sample size are relatively straightforward, variance changes resulting from variable coherence of the individual time-series are more complex to understand and thus address. Variable coherence among tree-ring series could be due to changes in the underlying climate system, and if so should presumably be preserved in the underlying dataset. However, inhomogeneities in tree-ring datasets that might occur e.g., in chronologies composed of samples from living trees linked together with relict / historical samples from more dissimilar (and/or unknown) environments should presumably be corrected. Similarly, changes in climate signal and coherence with tree age could result in variance artifacts in resulting chronologies. In such cases time-dependent estimates of \({\overline{{\varvec{r}}} }_{{\varvec{e}}{\varvec{f}}{\varvec{f}}}\) say computed in moving window (e.g., Frank et al. 2007) can be employed as included in more recent versions of ARSTAN (Cook and Krusic 2005).

It can be demonstrated that the variance structure of a tree-ring chronology is less dependent upon changes in sample size when \({\overline{{\varvec{r}}} }_{{\varvec{e}}{\varvec{f}}{\varvec{f}}}\) is high. And similarly, changes in sample size are more impactful to the variance at lower samples sizes. With specific reference to tree-ring stable isotope time-series, the general high correlation amongst series (Hartl-Meier et al. 2014) mitigates sample size variance artifacts. Conversely, the generally modest sample sizes for most tree-ring isotope studies to date are conductive to variance artifacts. Such variance artifacts can be mitigated by selecting for measurement a constant number of samples, or employing theoretically-based (see above) or empirically-based (Shiyatov et al. 1990) methods to correct for variance artifacts. Such considerations will be increasingly important to producers and analysts of tree-ring isotope datasets as the prevalence of variable sample replications increases as measurements are performed on all available tree-ring samples rather than small subsets. Most broadly, potential variance artifacts should be kept in mind in all scientific disciplines, including dendrochronology, where data quantities and qualities potentially change over time and space.

6 Conclusions

This chapter has reviewed foundations of dendrochronology with perhaps greatest relevance for an audience interested in the intersection of stable isotopes and tree-rings. The chapter began with a fairly broad overview of some of the fundamentals of tree-ring formation and wood anatomy, and then comprehensively outlined the diversity of research topics that have been addressed with a broad range of tree-ring parameters. The role of crossdating is highlighted as the key foundation for dendrochronology—namely to ensure that each and every ring is confidently assigned to a correct (and often calendar) year. For readers with a background in tree-ring width measurements, some of the long-standing paradigms that are being challenged with isotope data are perhaps of particular interest. This includes a discussion of the long-standing practices in selecting tree-ring sites towards ecologically extreme sites to obtain a strong common signal. There is now abundant evident that the stable isotope parameters in tree-rings carry an exceptionally strong signal where tree-ring width data do not. Such findings are opening up new possibilities in all domains of tree-ring research from crossdating and dendroarchaeology to dendroecology and climatology. Moreover, the role of stable isotope data, as well as quantitative wood anatomical data, in mechanistic / ecophysiological applications is highlighted as a research direction with excellent past work as well as substantial potential for further research. For readers that are more familiar with stable isotope measurements, and are part of the forefront in developing well-replicated tree-ring isotope datasets, the sections on chronology development, dendrochronological statistics, and characteristics of times-series may be most useful. Crosscutting issues in this chapter include discussions of age-size related trends in tree-ring data, and recommendations on how such age-size related trends might be characterized and handled for stable isotope datasets. The importance of producing robust well replicated tree-ring datasets is highlighted, and for which the signal and noise, variance, and chronology confidence are well understood and characterized across all time scales. Happy tree-ringing.