Late Weichselian varve archives re-explored to assess proglacial sedimentary chronologies using the principles of tree-ring analysis

Abstract

Glaciogenic varves are formed annually in proglacial basins and provide detailed geochronological archives about the dynamics of deglaciation. Varve chronologies are constructed from multiple varve series measured at individual localities. Temporal connection between the series is traditionally achieved by visual comparison of varve-thickness variations. This connection method resembles that of dendrochronological cross-dating. By contrast, dendrochronological correlation proceeds through the routine of statistical (along with visual) comparisons between the tree-ring series, providing quantification of the dating results, their veracity and robustness. Here we demonstrate the advantages and convenience of applying tree-ring statistical methods to varve records, using an extensive varve archive originating from southern Finland. The data series of individual varve-thickness diagrams were metamorphosed by detrending, prewhitening and averaging, that is, by the methods adopted from tree-ring science. Subsequently, it was seen that the series having 80 or more varves have a higher potential to be unambiguously cross-dated than shorter series. The varve-thickness diagrams could be categorized into two types, those having regional chronological importance and those showing only subregional (distance < 20 km) correlativity. The varves of former type ought to be regarded as primary constituents of the geochronology. The varves of latter type have low geochronological validity although they may bear local sedimentological importance. The findings have the potential to modernize the science of varve-based geochronologies.

Keywords

deglaciation dendrochronology Finland geochronology glaciology palaeoenvironmental reconstruction Salpausselkä Scandinavian Ice Sheet sedimentology varves

I Introduction

Glaciogenic varves are annually laminated sediment sequences displaying rhythmic variations of clay, silt and fine sand. They started to attract Nordic geologists by the late 19th and early 20th centuries. In Sweden, De Geer (1884–1885) first presented the possibility for varve measurements and geochronological correlations. The methods established by De Geer (1912, 1940) correlate varve records from different localities by means of diagrams, showing the thicknesses of consecutive laminae, and comparing the patterns of varve-thickness variations visually. In addition, comparisons of colour and lithological changes in the varved sediments were used to elaborate the stratigraphic interpretations (De Geer, 1940). Thereafter, the examination of the late Weichselian and early Holocene varves has continued and advanced in Sweden (e.g. Brunnberg, 1995; Cato, 1987; Holmquist and Wohlfarth, 1998; Johnson and Ståhl, 2010; Nilsson, 1968, Strömberg, 1983, 1985, 1989; Wohlfarth et al., 1998) and it has been suggested that this composite sedimentary record could span roughly over the past 13,300 years (Wohlfarth et al., 1995).

The late Weichselian depositional conditions were largely similar in lacustrine and marine environments of Finland where varved sediments have been documented in a number of outcrops and brickyards. The most comprehensive studies on the varves in the region were compiled already by Sauramo (1918, 1923) who made use of these archives to build a varve chronology that comprised 152 local varve-thickness diagrams from the southern part of Finland. The published record of Sauramo (1918, 1923) spanned 2215 varves and was expected to provide a detailed deglaciation chronology for the last recession of continental ice (i.e. the Scandinavian Ice Sheet) before, during and after the formation of the Salpauselkä end moraines. These varves were deposited in a proglacial basin with a water depth varying between 50 and 150 metres, and with the distance to the receding ice margin being not more than 20 km during deposition (Figure 1). Later examinations have agreed with the chronology of Sauramo (1918, 1923) which occupies a timespan between approximately 13,100 and 10,800 calendar years ago (Saarnisto and Saarinen, 2001; Wohlfarth et al., 2008). Unlike its Swedish counterpart (Cato, 1987), the lateglacial varve record of Finland (Sauramo, 1918, 1923) has not been linked to the present-day contemporary varve deposition.

Figure 1.

Varve sites (Sauramo, 1918) with a palaeogeographical reconstruction of the deglacial features in the area. The numbers refer to Sauramo’s (1918) sites.

Subannual and interannual changes in the sediment output and deposition form the basis for the interpretation of varves and their fine-scale chronological studies. Essentially, the subannual rhythm drives varve formation on annual basis and the interannual variability enables the comparisons between two or more records. Varve chronologies are constructed as assemblages of varve-thickness series measured at multiple localities. Traditionally, the connection (that is, the temporal linkage through conspicuous synchrony) between varve records is made on visual basis by comparing the interannual thicknesses of varves. In so doing, the geochronologists relying on varves as basis of their deglaciation studies have utilized a method that is not unlike the procedures that are globally employed by dendrochronologists to date tree-ring series. Dendrochronological dating approaches seek to systematically correlate the growth patterns observed as variation of wide and narrow rings (Stokes and Smiley, 1968; Yamaguchi, 1991). However, it is noteworthy that tree-ring series are additionally compared using statistical measures as a routine of dendrochronological data processing (Baillie and Pilcher, 1973; Fritts, 1963; Grissino-Mayer, 2001; Holmes, 1983). Combined, the approaches constitute the methodological framework for dendrochronological cross-dating (Fritts, 1976; Schweingruber, 1988).

The statistical applications of dendrochronology enable objective assessment and quantification. While the addition of statistical approaches brings clear benefits to simple visual dating, they have not become tools of geochronological studies based on glaciolacustrine varve thickness. The main aim of this paper is to explore the possibilities of applying the dendrochronologically developed dating methods to lateglacial varve data. An emphasis is also given for the application of methods aiming at reducing noise in varve series prior to statistical cross-dating. These are the methods of detrending and prewhitening that have been accepted as more or less mandatory steps in dendrochronological analysis (Cook, 1985; Fritts, 1976). We demonstrate that these methods make it possible to construct varve chronologies similar to tree-ring chronologies by averaging varve series into summary chronologies comprising a multitude of detrended and prewhitened varve series. We also show how these methods maximize the common sedimentary signal between the varve records while minimizing the risks of spurious dating.

II Material and methods

1 Spatial and temporal setting

In southern Finland, the deglaciation pattern is particularly evidenced by the large uniform 1st and 2nd Salpausselkä end moraines formed during the late Weichselian (Figure 1). They constitute two ridge-like arcs indicating the terminal geometry of the ice-lobes associated with the Younger Dryas cold event (Donner, 1995; Rainio et al., 1995). At that time the glacier front stopped retreating and the glacier margin was reorganized to adapt the Younger Dryas flow regime. According to recent geochronological schemes, this occurred at about 12.6–11.5 ka BP (Saarnisto and Saarinen, 2001; Wohlfarth et al., 2008). The exposure ages by ¹⁰Be method suggest an estimate of 12,500 ± 700 years BP for 1st Salpausselkä end moraines (Rinterknecht et al., 2006; Tschudi et al., 2000).

Sauramo (1918, 1920) published varve-thickness diagrams from 60 localities in southern Finland. The diagrams were dated on the basis dated by varve thickness (i.e. visual connection), colour and lithological changes. In the original publications, the zero-year of the chronology was set to the varve expected to have been deposited at the end of the 2nd Salpausselkä phase. With regards to the history of the Baltic Sea, the zero-year would thus denote the draining of the Baltic Ice Lake to the level of the Yoldia Sea. The varves below the (presumed) drainage horizon were aligned according to negative years whereby the oldest (that is, stratigraphically lowest) varve was assigned to year –1484 (Sauramo, 1918). More recent studies have suggested dating problems relating to the varves that were deposited after the glacier withdrawal from the Salpausselkä position (Niemelä, 1971; Strömberg, 1990, 2005).

Prior to examining these problems, it would be beneficial to better understand the methodological requirements necessary to adapt the tree-ring-counting techniques to varve dating. To study the potential of the varve-correlation method proposed here, a data set was created by digitizing all the diagrammed varve counts south of the 1st Salpausselkä as presented in Sauramo (1918) (Figure 1). The data of this study spans 867 years including a subset of 47 varve localities (Figure 2). Comparisons of varve-thickness diagrams from Finland and Sweden have previously suggested several possible temporal connections between the chronologies (De Geer, 1940; Sauramo, 1926; Schove, 1971; Strömberg, 1990, 2005). According to a recent suggestion, the temporally floating zero-year of Sauramo (1918, 1923) may correlate to the year of 8693 bc. (Strömberg, 1990). Consequently, the chronology examined here would represent the years between 10177 and 9311 bc, given that the Swedish varve dating is absolute in terms of calendar years. However, for clarity, the original varve year scale (Sauramo, 1918, 1923) was maintained in this study. This approach would additionally enable direct comparisons to earlier studies. In all, the data examined contains 3710 thickness measurements of individual varves. Geographically, the data can be divided into western and eastern subregions (Figures 1 and 2) comprised of 26 and 21 varve-thickness diagrams, respectively. Within this assemblage, the diagrams comprised, on average, 79 varves. Eleven diagrams contain more than one hundred varves, whereas 32 diagrams have at least 50 varves; the longest and shortest diagrams totalled 291 and 16 varves, respectively.

Figure 2.

Temporal association between the varve series (bars) according to the original dating by Sauramo (1918). The varve-thickness diagrams from sites 2 and 33 were not presented graphically in the original study and are thus not included in this study. The data represents years between approximately 10,200 and 9300 bc.

2 Statistical transformation of varve-series data

A ubiquitous feature present in the late Weichselian varve-thickness diagrams is the trend in thickness from proximal to distal varves (Hang, 1997, 2003; Sauramo, 1918, 1923; Strömberg, 1990; Wohlfarth et al., 1998). That is, an exponential diminution in varve thickness is evident towards the distal end of the diagram. An approach, to quantitatively characterize this trend, is to fit a negative exponential curve, such as that of Fritts et al. (1969), to each varve-thickness diagram and adopt the regression constants as descriptive variables for each trend. Here we give this equation to determine the modelled lamina width (w_m ) in a local varve number of the diagram (v) defined as

w_{m} (v) = p_{e}^{- c v} + d

where p controls the initial height of the curve (thus approximating the thickness of proximal varves), c determines the curvature (i.e. concavity) of the trend and d exhibits the terminal height of the curve (thus approximating the thickness of the distal varves) (see also Table 1). The behaviour of the function is exemplified in the case of one locality in Figure 3a. Alternatively, linear regression was used to model the trend for the varve-thickness diagrams without curvature in their trend. Importantly, the trend model quantifies large portions of the total variability in varve thickness, owing to increasing distance between sediment output and deposition as a function of receding ice margin (De Geer, 1912, 1940; Sauramo, 1923) and rising of the water level in the ice lake (Donner, 1995). Both of these factors tend to widen the accommodation space and, consequently lead to thinner varves. Therefore the existence of the trend in the diagrams is due to factors inherent to progressing deglaciation, which would be supported by the general similarity between the trends in different diagrams from different regions (Hang, 1997, 2003; Sauramo, 1918, 1923; Strömberg, 1990; Wohlfarth et al., 1998). Accordingly, the formula (equation 1) would in large part separate these influences from those external to ice recession and base level rise; likewise, the mathematical elimination of the trend (i.e. detrending) is expected to result in varve data independent of a recession signal.

Figure 3.

(a) Varve diagram from locality 41 (black line) and the modelled decline in varve thickness (grey line; equation 1). (b) Detrending the varve-thickness diagram transformed the original measurements into a dimensionless series of varve-thickness variability (grey line) that was further prewhitened (black line).

Table 1.

A short glossary of the nomenclature

Symbol/term	Definition
c	Parameter estimating concavity of the decay model (equation 1)
d	Parameter estimating width of the distal varves by the decay model (equation 1)
D	Distance between varve localities (dimension: km)
Local chronology	Varve chronology comprising a number of cross-dated varve series with D < 20 (dimensionless)
i	Ratio between w_o and w_m (dimensionless)
N	Temporal overlap between two varve records
p	Parameter estimating width of the proximal varves by the decay model (equation 1)
r_local	Pearson correlation between two varve series with D < 20
R_local	Pearson correlation between a varve series and local chronology
r_regional	Pearson correlation between two varve series, thus independent of D
R_regional	Pearson correlation between a varve series and regional chronology
Regional chronology	Varve chronology comprising all cross-dated varve series, thus independent of D (dimensionless)
t_local	Student’s t; calculated as a function of N and R_local
t_regional	Student’s t; calculated as a function of N and R_regional
v	Local varve year in each locality (v = 1, 2, 3…) (equation 1)
Varve chronology	Local chronology or regional chronology (dimensionless)
Varve diagram	Original varve measurement data (dimension: cm)
Varve record	Varve series or varve chronology (dimensionless)
Varve series	Detrended (or detrended and prewhitened) varve data of one locality (dimensionless)
w_m	Estimated width of individual varve by the model (equations 1 and 2) (dimension: cm)
w_o	Observed (measured) width of individual varve (equation 2) (dimension: cm)

In principle the separation of the trend and variations of shorter term is simply to divide or subtract the value of the actual varve thickness from that of the fitted curve, year by year (equation 1). The original varve data are known to be heteroscedastic (Holmquist and Wohlfarth, 1998) – that is, the means and variances of consecutive varve thicknesses are not independent. Therefore, it would be advantageous to divide rather than subtract since this would transfer the resulting data into a stationary series (Cook and Peters, 1997; Helama et al., 2004; Matalas, 1962) thus simultaneously removing the potential diagram-long trend in variance. Here, the model with the best fit was chosen for each series (equation 1 or linear regression). Thereafter, the varve-thickness diagrams were transformed by computing a dimensionless ratio between the value of measured varve thickness and the model (Figure 3b). In so doing, the dimensionless varve series could be formed as a series of ‘indices’ (i) computed as ratios between the observed (w_o ) and modelled (w_m ) varve thicknesses by:

i (v) = \frac{w_{o}}{m_{m}}

Apart from trend, many types of natural time series are typically autocorrelated. This is known to be the case of varve-thickness records as well (Young et al., 2000). The presence of autocorrelation indicates that the thicknesses of consecutive varves may not be independent, but serially dependent on each other. A principal constituent of this autocorrelation likely originates from climate, which often shows persistence via fluctuations and transient trends (Karl, 1988; Trenberth, 1984). Another source of autocorrelation could originate from the melting and circulation processes in the sedimentary basin (Sauramo, 1923). In this sense, autocorrelation may be linked to depositional mechanisms causing such inertia (Young et al., 2000). The level of serial dependence was determined here for each series by calculating autocorrelations and partial autocorrelations at lags 1 to 25.

The presence of serial autocorrelation may hamper temporal analyses (Bartlett, 1935; Quenouille, 1952; Yule, 1926). To remove the autocorrelation, the detrended varve series were entered into the autoregressive-moving average (ARMA) modelling (referred to as ‘prewhitening’). The ARMA models of Box and Jenkins (1970) determine the underlying structure of mathematical persistence in time series. Residuals from ARMA models are thus series that have practically lost their autocorrelation. The order of ARMA model used was determined individually for each index series using Akaike’s (1974) information criteria. These prewhitened series are used here to re-examine the sedimentary signals in the varve series.

3 Assessing the chronological control

Construction of varve chronologies presumes temporal overlap and similarly constrained sedimentary variations, among two or more series, from different localities. In statistical terms, the number of overlapping varves determines the sample size, N, for each comparison. The similarity in sedimentary variations, on the other hand, could be quantified using the Pearson product-moment correlation coefficient (r) that describes the linear relationship between two series. Compared to variations of a single series, the variations in the mean of several series are more reliable (e.g. Wigley et al., 1984) and, in dendrochronological dating, it is common to calculate correlations between an undated series and the mean of already cross-dated series (e.g. Holmes, 1983). While a single series represents site-specific sedimentary variations, the mean of several series (i.e. chronology) can represent either local or regional domains, depending on the areal extent of the provenance. Here we aimed at detailing the geochronological opportunities of varve dating by constructing varve chronologies of spatially restricted and regionally representative data, and the corresponding series-chronology correlations were denoted as R_local and R_regional , respectively (see also Table 1).

Combinations of these measures, N and R_local or R_regional constitute yet another statistical measure that has often been used in dendrochronological studies, the t-value (Baillie and Pilcher, 1973). This value is defined as:

t_{r e g i o n a l} = \frac{R_{r e g i o n a l \sqrt{(N - 2)}}}{\sqrt{(1 - R_{r e g i o n a l}^{2})}}

where t_regional is calculated using correlations between the single varve series and regionally representative varve chronology. Alternatively, the t_local could be calculated using R_local instead of R_regional . The Student’s t provides a measure of probability of the observed correlation to have arisen by chance. A t-value of 3.5 was originally suggested by Baillie and Pilcher (1973) as a minimum for an acceptable dendrochronological synchrony. These statistics measuring the level of synchrony, R_local , R_regional , t_local and t_regional , were used here to determine the agreement between the sedimentary variations and thus of the varve dating. In order to avoid spurious synchrony estimates, correlations were calculated for pairs of series with N ≥ 15 only.

The original varve data (Sauramo, 1918, 1923) were re-examined here using the detrended and prewhitened series. (1) Each series was correlated with all other series in their suggested temporal positions. (2) Each sample was lagged forward and backward in time to determine whether offsetting the time series would yield higher visual and statistical correlation. In the case of higher correlation in a new position, the number of years lagged was taken as an indication of the number of offsetting varves. (3) The varve series was divided into overlapping segments before the lag analysis. Identification of the segment in which a correlation substantially drops was used to locate the year of a dating error. Following the suggestion of Douglass (1941) for dendrochronological analyses, the final decision was made by visually inspecting each sample series against the other series.

Once the temporal agreement between two or more series was determined, the varve series could be accepted into the chronology that was computed as an arithmetic mean series. Subsequently, the t-value was calculated between the individual series and the average of all other series.

III Results

1 Detrending

The mean varve thickness was about 1 centimetre and the first-order autocorrelation of the varve-thickness diagrams averaged 0.570. Progressive decreases in annual sedimentation, from proximal to distal varves, were seen as pervasive feature of the diagrams. All except one varve-thickness diagram (locality 22) exhibited a negative trend in thickness. Thirty-seven out of 47 varve series were modelled using a concave trend (equation 1). Linear regression was determined to be a more suitable trend for modelling the remaining series.

2 Prewhitening

Positive correlations were in general found between the series in their presupposed temporal positions as suggested by visual cross-dating (i.e. those of Sauramo, 1918). Statistically speaking, the mean inter-series correlation was 0.374. After detrending, the first-order autocorrelation of the varve series decreased to a mean of 0.287. Following prewhitening, the mean inter-series correlation ascended to 0.439.

3 Geographical assessments

Spatial comparison between the series revealed systematic relations between the varve-thickness variations and the distance (D) between the sites of deposition. In regional view, the correlations between the series were generally higher where the distance between the localities was shorter (Figure 4a). The correlation analysis could be appropriately divided into long (here D > 30 km) and short (here D < 20 km) distances. On an average, the correlation between localities was 0.501 and 0.384 on short and long distances, respectively. Moreover, the relationships between D and r could thus be analysed separately on corresponding spatial scales, as shown in Figures 4b and 4c. Consistent with the regional view, this analysis showed that the correlations between the localities were, in general, in decline with increasing distance with a weaker correlation-distance relation subregionally.

Figure 4.

Distance-dependent correlations between the varve series. Assessment of correlations on (a) regional, (b) subregional and (c) local scales. Spatial assessment is indicated by the slope of the trend line and quantified further by Pearson correlation (r). Equation of linear regression quantifies the linear dependence between the correlativity (r) and distance (D).

4 Local versus regional sedimentary information

Next, the correlations were calculated between the varve series on local (r_local ) and regional scales (r_regional ) and between a varve series and the arithmetic mean of other series on local (R_local ) and regional scales (R_regional ). Here, the local values were computed for pairs of varve records with D < 20 km only but the regional estimates were derived with independence of D (see also Table 1). In majority of cases, the correlations were higher when calculated on local (i.e. using r_local or R_local ) rather than regional scales (i.e. using r_regional or R_regional ) (Figure 5a). Thirty-three and 22 out of 47 varve series showed higher r_local and R_local , respectively, in comparison to r_regional or R_regional . In terms of geochronology, this would certify that it is beneficial to compare varve series from nearby localities, rather than using extended spatial scales, for dating purposes. In addition, it was evident that the calculation of correlations between the varve series and the varve chronology (i.e. using R_local or R_regional ), rather than between two or more varves series (i.e. using r_local or r_regional ), yielded considerably higher correlations (Figure 5b). Moreover, this result was consistent both on local and regional scales. It is notable that R_local and R_regional exhibited equal skill since the 47 series showed averages of R_local and R_regional as similar as 0.555 and 0.557.

Figure 5.

Comparison of varve correlativity (a) between local and regional scales and (b) between the locality-specific and locally and regionally averaged varve variability (see also Table 1).

5 Assessment of visual dating

Overall, the t_local and t_regional averaged 5.61 and 5.95, respectively, with values exceeding the level of 3.5 in the case of 36 and 37 series (Figure 6). Thirty-four series showed t_local and t_regional both exceeding the predetermined t-value level whereas eight series had both t_local and t_regional below that level. These series originated from localities 9, 17, 22, 28, 35, 39, 45 and 48. These series were in general characterized by low correlativity in comparison to remaining series, but especially by overall shortness of the series (Table 2).

Figure 6.

Assessment of the varve series as guided by t-values. Horizontal line shows the predetermined level of 3.5. Numbering of the localities follows the charts by Sauramo (1918).

Table 2.

Comparison of varve series showing unambiguous (t_local > 3.5 or t_regional > 3.5) and ambiguous (t_local < 3.5 and t_regional < 3.5) datings in their presupposed temporal position. Series in these two groups were characterized here by the number of series in each group, their mean lengths (varve years), and averaged series-series (r) and series-chronology (R) correlations calculated separately on local and regional spatial scales. The standard deviations of the estimates are given in brackets.

	Number	Length	r_local	r_regional	R_local	R_regional
Unambiguous series	39	85.4 (53.0)	0.537 (0.127)	0.476 (0.108)	0.577 (0.108)	0.589 (0.109)
Ambiguous series	8	32.4 (14.4)	0.402 (0.098)	0.318 (0.169)	0.431 (0.138)	0.374 (0.179)

In addition, there were two series showing t_local above and t_regional below the acceptable level (Figure 6). This would not appear surprising due to generally higher correlativity of the series at short distances (Figure 4). However, there were three series showing t_regional above and t_local below the same level. These were the series from localities 1, 16 and 21, which all originate from the eastern subregion which contains fewer localities (and thus varve series) than the western subregion. Moreover, the series 1 and 21 originate from the southern and northern ends of the eastern subregion (Figure 1) with restricted temporal overlap with series from adjoining localities. While this would exemplify the general difficulty of dating the varve series near the ends of the chronology, it also becomes evident that the arising difficulty could be surmounted by using t_regional instead of t_local .

6 Stripped chronology

The eight series with ambiguous dating assessments were next excluded from the chronology. The resulting experimental, stripped, chronology thus contained 21 and 18 series from western and eastern subregions (Figure 7). The stripping resulted in increased series-chronology correlations with 39 series having R_local and R_regional average of 0.589 and 0.594, respectively. The improved correlativity also increased with t_local and t_regional to 6.38 and 6.74, respectively.

Figure 7.

Temporal distribution of varve data originating from localities of western and eastern subregions. The data represent years between approximately 10,200 and 9300 bc.

An additional trial was performed to find out if any of the remaining 39 series could be alternatively dated with improved statistics and to detect possible missing or falsely added varves. It was found that all except two varve series bore their highest R_regional in their predetermined positions. These were the two series, from localities 40 and 44, of which t-values barely exceeded the level of 3.5 in their presupposed positions (Figure 6). In the case of the two series the original datings were accepted here due to the reliable appearance in their visual dating and geological consideration by Sauramo (1918). Moreover, the rest of the varve series showed highest correlations (and thus t-values) for their presupposed position (Figure 8, a and b) with fluctuating but positive correlations over different segments (Figure 8c).

Figure 8.

(a) Dating process of an individual varve series against the stripped varve chronology. In this case, the varve series shows strong correlation against the varve chronology in its initial temporal position. (b) The correlation in this position was considerably higher than in any other temporal placement lagging or leading the initial position by 250 years. (c) Running correlations using a 50-year window showed strong correlativity through time at the same position. This locality was selected as a typical series for illustrating the cross-dating.

IV Discussion

Both varves and tree-rings provide incremental archives of palaeoenvironmental changes spanning annual to millennial timescales. The two types of geochronologies are frequently examined in the literature but the disciplines of varve and tree-ring sciences have thus far evolved independently to a large degree. This study initiated from the premise that the time-series analysis methods developed in the context of the tree-ring science could benefit opportunities for varve research. Notwithstanding the major contrast between sediment deposition and wood cell growth, varve and tree-ring data do portray an array of similarities that suggest the applicability of detrending, prewhitening and the construction of mean chronologies for varve data. Subsequently, the cross-dating approaches that were originally developed for the purposes of dendrochronology were applied here for the varve-thickness data. Previously De Geer (1940) had provided a comprehensive list of 18 potential reasons that could result in inconsistent varve-thickness variations. Some of the issues were physical, originating from glaciological or sedimentological factors, some of them stemming from observational errors. Digitizing the varve thicknesses from original publications may represent an additional source of uncertainty. Regardless of their initial origins or causality, any of these elements could be referred to as noise, in terms of geochronological dating. Consequently, a failure to take these factors into account could critically lower the possibility of successful geochronological comparisons. In the subsequent sections, we aim at further exploring the issues of chronological signal and noise in the compound context of varve and tree-ring literature.

1 Dependence of sequential increments

The raw varve-thickness diagrams showed elevated autocorrelations and trends of diminishing thickness from their proximal to distal increments; that is, the varve-thickness variations portrayed persistence on short and long timescales, respectively. In other words, the amount of deposition of clay and silt particles in a given year was found to be dependent on the sedimentation of suspended matter in preceding and subsequent years. Importantly, these findings were not restricted to our materials, but varve data elsewhere have demonstrated similarly negative trends and higher autocorrelation (Hang, 1997, 2003; Wohlfarth et al., 1998; Young et al., 2000).

The existence of the trend in the varve-thickness variations has previously been explained by the increasing distance between the glacier margin and the site of deposition (De Geer, 1912, 1940; Niemelä, 1971; Rainio, 1993; Sauramo, 1918, 1923). Here, 37 of 47 varve-thickness diagrams showed an exponential decline in varve thickness with time. The trend probably reflects deposition from ice proximal to more distal environments. It could be assumed that the increasing distance to the ice margin has relatively monotonic effect on sedimentation whether the ice margin is only a few kilometres or tens of kilometres away, as long as glacier meltwaters are feeding sediment into the basin. Similar change is observable in the varve-thickness diagrams showing monotone trend of thickness diminishment subsequent to the steeper proximal part of the trend (Figure 3).

Yet we note that the trend and its shape may become altered by variations of the water level caused by changing proglacial hydrology or by subaqueous mudslides generated by earthquakes (Gruszka, 2007) or just by changing sedimentation dynamics. A rising trend in the base level means a steadily growing accommodation space, leading to a predominant deposition of distal bottom plain varves with an upward thinning trend in stratigraphy. A base-level fall, generally, could increase erosion, redeposition and local sediment slumping and sliding as sediments deposit in more proximal setting mostly by traction or turbidity currents (Gruszka, 2007). All the varve series adapted to this study are older than the Baltic Ice Lake drainage but we think these phenomena could generate small hiatuses in varve records, thus hampering the geochronological correlativity between varve records, in other regions and varve records. The effect of changing base level has not thus far been analysed from the Finnish varve series, but an enhanced understanding of this process could likely pave the way for improved geochronological models (Hyttinen et al., 2011).

An explanation for the autocorrelation, on the other hand, may to a degree originate from climate which is known to show persistence caused by multi-annual anomalies and transient trends (Karl, 1988; Trenberth, 1984). Apart from long- and mid-term trends, one more factor involved in the short-term persistence in the varve data may result from glacier dynamics and circulation processes influencing the sedimentation in the basin, as already noted by Sauramo (1923). More recently, Young et al. (2000) put forward an explanatory mechanism likewise internal to the depositional system, by expecting multi-annual residence time for depositing material of the finest grain size. Here, we postulate that this applies especially to ice distal environment where suspension is main sediment transport mechanism.

It is noteworthy that the negative trends and positive autocorrelations also characterize tree-ring series, which are typically wider near the pith and exponentially decrease toward the bark (Fritts, 1976). Likewise in dendrochronology, the trend of diminishing ring width is thought to arise from the geometric constraint of adding an annual volume of biomass to a stem of a tree with increasing radius (Cook, 1990) with environmental controls on the trend shape (Helama et al., 2005b; Naurzbaev et al., 2004; Nicault et al., 2010; Spurk et al., 2002). Moreover, the autocorrelations of tree-ring width data can be markedly higher than those observed in climate records (Helama et al., 2009), indicating the effects of physiological preconditioning such as accumulation and storage of food and other substances within trees and their lingering effects on current growth from previous conditions (Fritts, 1976). As a highly topical coincidence, both the trend and autocorrelation thus appear primarily resulting from factors internal to the sedimentological or biological systems. As a dramatic difference, on the contrary, the dendrochronologists recommend (Cook, 1985; Grissino-Mayer et al., 2010; Monserud, 1986; Yamaguchi, 1986) removing the biological variations prior to the actual cross-dating procedure whereas the varve studies typically compare all types of thickness variations apparent in the diagrams regardless of their assumed origin.

2 Towards unbiased assessments

Following the dendrochronological practice of removing the biological variations present in the ring-width data, the varve-thickness diagrams were treated here using the adopted methods of detrending and prewhitening. Ultimately, the cross-dating procedure is to seek for common variations between the varve records. First, virtually all initial varve-thickness diagrams exhibited such a common variation in the form of their negative trends and, consequently, any pair of varve-thickness diagrams could thus be doomed to exhibit predominantly positive statistical correlations regardless of their geochronological position (see Grissino-Mayer et al., 2010). Consequently, these adverse effects of the trends were eliminated by detrending the initial varve-thickness diagrams.

Second, the trouble that autocorrelated series exhibit false synchronies more frequently than the series of random observations (Bartlett, 1935; Quenouille, 1952; Yule, 1926) was avoided by prewhitening. In practice, the removal of both the trend and the autocorrelation results in the isolation of high-frequency variations that is thus highly advantageous in the cross-dating procedure (Wigley et al., 1987). The benefits of this isolation has recently been exhaustively demonstrated by Grissino-Mayer et al. (2010) with the concern for the noisiness of the cross-dating attempts using non-detrended and non-prewhitened tree-ring data. Yet another example was demonstrated by Yamaguchi (1986) who tree-ring-dated a Douglas fir stump with 290 rings exhumed from a pyroclastic-flow deposit in association with the eruptions of Mount St Helens in Washington state. The dating was achieved using the detrended and prewhitened tree-ring series from the stump, correlated with the reference chronology. In our study, the corresponding benefit was also obtained as an increased correlativity between the visually dated varve series due to prewhitening. Evidently, the removal of autocorrelation not only decreased the possibility for flawed datings, in theory, but also increased the common chronological signal, in practice.

Third, the effective sample size (pertaining to the temporal extent) can be considerably less than the actual diagram length due purely to autocorrelation. Using the formula of Quenouille (1952), and adopting the average first-order autocorrelation of 0.570, the average diagram length of 79 varves would undergo a drop to an effective varve number of 40. Halving the size of information could be seen as deleterious for cross-dating procedure as the difficulty of dating particularly the short series was demonstrated (Table 2).

3 Construction of mean chronology

It was also demonstrated that using R_regional could be advantageous owing to its ability to capture synchronous sedimentary variations better than the other similar estimates (R_local , r_regional or r_local ). An underlying difference between the used estimates of statistical agreement is that R_regional is calculated between the series under investigation and the mean of all other series. It is arguable that the mean chronology is able to record the common sedimentary variability consistently, by cancelling out the variations that are present in individual series only, thus providing a reliable backbone for varve-dating purposes (Fritts, 1976). Such variations, which cannot be traced from a larger set of series, could presumably result from local depositional peculiarities or random measurement error (De Geer, 1940). These variations could be regarded as noise (unwanted variability), in terms of chronology construction, as they can lessen correlations between records in general. An additional benefit of using R_regional is that the mean chronology provides a continuous reference against which individual series can be compared whereas, in the case of r_local or r_regional , the chronological comparisons are made using fragmentary data of individual varve series. In the case of the present sample, the mean chronology was calculated as an average of varve series from unrestricted provenance, and R_regional thus provided with a mixture of local and regional sedimentary variations. This would appear as a complementary benefit of using R_regional as a reference value for dating since some of the series showed particularly consistent association with the regional sedimentary variations. For example, the locality 16 yielded R_local = 0.227 and R_regional = 0.494. While the specific characteristics in the sedimentary variations cannot be known a priori, the usage of R_regional would compromise between the very local and broadly spread, spatially large-scale, sedimentary variations thus providing robust estimate for the ubiquitous sedimentary signal.

The benefit of constructing the mean chronology and using it for cross-dating using tree-ring statistical techniques is evident. We note that a somewhat similar approach was already carried out by Antevs (1925a, 1928), by constructing what he called ‘normal curves’, when studying the ice-retreat and deglaciation history of northeastern North America. The normal curves were the mean records of several varve-thickness diagrams. They were calculated subsequent to cross-dating which was practised by the methods of De Geer (1912). The averaging procedure of Antevs (1925a, 1928) excluded the diagrams with anomalous varve thickness or poor agreement in the curve shape. Similar to the methods of De Geer (1912), the procedure did not aim at detrending the varve-thickness diagrams but it appears that Antevs (1925a, 1928) indeed hesitated to average proximal and distal varves. It is conceivable that the innovation of averaging data originated from his wide scientific background as Antevs (1925b, 1925c) had also published tree-ring studies. He had even developed a new tree-ring detrending method (Antevs, 1925c), which he did not apply in his varve studies.

Moreover, Wohlfarth et al. (1998) calculated a ‘filtered’ mean varve-thickness diagram for southern Sweden by first omitting the thick proximal and drainage varves and excluding fully the diagrams with anomalous overall varve thicknesses. In contrary to the present study, the resulting mean chronology of Wohlfarth et al. (1998) was used for interpreting the alternating periods of melting, rather than to improve dating veracity. We note that this method is rather similar to the palaeoclimatic approach of LaMarche (1974) who examined bristlecone pine tree-rings grown at the upper tree-line in the White Mountains of eastern California. That is, LaMarche (1974) omitted ring-width measurements from near the centre of tree stem to exclude the growth variations due to plant ontogeny in the resulting average chronology. As later noted by Cook et al. (1995), this method would, however, require rigorous testing for the assumption that the series do not have any trends apart from the omitted sections. Moreover, the exclusion of the thick proximal varves would likely shorten the series unreasonably (see Figure 3). The latter issue would become crucial considering the importance of series length for the successful cross-dating, an issue that was explained above in more detail. The exclusion of parts of the series would also result in unreasonable decrease of chronology sample size (i.e. replication) or even chronological gaps.

4 Correlativity across local and regional spatial scales

Apart from temporal correlations between the varve records, an additional aspect, crucial in chronology development, is the spatial character in the correlativity between the varve records. This correlativity was studied here using the prewhitened varve series because of their particular relevance for the cross-dating procedure. Consistently, the prewhitened tree-ring series have previously shown to provide more coherent patterns of spatial correlation than series of ring-width without prewhitening (Henttonen, 1984). Modelling the spatial aspects by the regression of the correlativity on distance (Figure 4) quantified the slope of diminishing spatial synchrony with longer distances. The general picture of this spatial character is reasonably similar to the view obtained from previous varve studies also indicating decreasing correlativity with increasing distance.

Historically, varve correlations were confirmed by comparisons between sites closer than approximately 5 km from each other (De Geer, 1912: Plate 2). Even then, the spatial confidence of correlativity was extended during the subsequent years of related research over global scales (e.g. De Geer, 1927). These overstatements were later moderated by more recent recommendations; for example, Strömberg (1983) suggested that the distance between separate varve measurements should not exceed 1 km. Subsequently, a slightly broader geographical restriction suggested comparing only the sites of varve measurements located no more than 5–10 km from each other (Holmquist and Wohlfarth, 1998). Using our regional model of spatial synchrony (Figure 4a) and parameterizing equation 3 using N = 79 reveals that the t-value drops below the level of 3.5 (Baillie and Pilcher, 1973) when the distance between the sites of deposition exceed approximately 40 km. It could thus be implied that the methods applied here made it possible to recognize varve connections over longer geographical distances than previously suggested by visual comparisons.

It should be additionally emphasized that a more consistent correlation is expected using averaged rather than individual varve (or tree-ring) records. Consequently, ameliorated t-values could be obtained for longer distances by comparison between independently constructed mean chronologies from distant (here D > 40 km) regions. Regarding dendrochronological estimations, for example, it has been shown that the mean chronologies constructed from Scots pine tree-ring widths can be synchronized within 800 km in northern Fennoscandia (Helama et al., 2005a; Schweingruber, 1988). These implications demonstrate the potentials resulting from averaging of the data over progressively larger spatial scales (local-to-regional) to constitute cross-dating over clearly extended distances.

5 External forcing

The fundamental issue in interpreting the varve synchrony is the origin of the observed variability. As previously alluded to, the variability in varve records may become simplified relative to the initial varve-thickness diagrams whose thicknesses result from a mixture of all factors influencing the final sedimentation in compound fashion. Briefly, this becomes possible as hypothesized effects of changing deposition mode, the retreating glacier margin, glaciolacustrine circulation dynamics and localized depositional processes are expectedly removed from the diagrams by detrending, prewhitening and averaging. As a result, the variations observed in the resulting varve records were expectedly dominated by signals from factors external to the aforementioned processes of deposition.

Topically to our results, De Geer (1940) already pointed towards the ‘biennic’ climatic fluctuations as factors behind observable varve synchrony. As discussed above, the depositional variability most closely regarded as biennial, or interannual, becomes critically enhanced by the chosen time-series method of our dendrochronological approach (Cook, 1985; Grissino-Mayer et al., 2010; Monserud, 1986; Yamaguchi, 1986). Combined interpretation of related sedimentological evidence (Leeman and Niessen, 1994; Sander et al., 2002; Tomkins et al., 2008) and the correlativity patterns obtained here (Figures 4 and 5) further confirm that the role of climate variability behind the temporal and spatial varve synchrony is relatively well established, and that clastic varves can be related with summer temperatures (e.g. Leeman and Niessen, 1994) and precipitation control (e.g. Desloges and Gilbert, 1994). Also, extreme flood events or debris flows can be seen in varve records as anomalously thick varves (e.g. Sander et al., 2002; Tomkins et al., 2008). To this end, the varve-thickness variations in our late Weichselian setting could be expected to reflect similar variations in discharge and summer climate.

In fact, the spatial pattern of declining varve correlativity (Figures 4 and 5) is parallel to the behaviour in the geographical synchrony of instrumentally observed climate variability (Koenig, 2002). As determined from weather observations, the spatial synchrony of air temperatures is known to decline at increasing distances, while correlations as high as 0.7 could be observed between meteorological stations located 1000 km from each other in the study region in modern times (Heino, 1994). Indirect climate series such as palaeoclimatic proxies as tree-rings are, however, known to exhibit spatial correlations with more rapid declines than actual climatic data (Holopainen and Helama, 2009; Rolland, 2002), this finding clearly agreeing with our results on the late Weichselian varves and their spatial connectedness. Interestingly, the slope of decline observed here regionally (Figure 4a) was identical to that observed previously for historical agricultural productivity of southern Finland that have peculiarly been dependent on the seasonal melting processes of the ice and snow layers during the Little Ice Age (Holopainen and Helama, 2009).

Last, we note that still the non-climatic factors influencing the spatial character of correlations could relate to the positions of feeding meltwater channels. That is, the probability of underflows is greater in the proximity of esker chains (see Figure 1). There is a possibility that the underflows may erode the deposited sediments causing local variations in varve-thickness diagrams. Topography could thus be considered as a factor causing local to subregional variance in varve deposition. The two studied transects of Sauramo (1918) differ in relation to esker chains (Figure 1). The eastern transect is in the proximity of an esker chain, but in the west the feeding meltwater streams have been at some distance. However, this did not seem to affect the correlativity of the series in our case (Figure 6).

6 Incremental agreements

The original varve connections were obtained by Sauramo (1918, 1923) on a visual basis. Here, we have focused on the power of statistical approaches, adopting them from the dendrochronological dating tradition. Meanwhile, the capability of mathematics to cross-date incremental series whether varves or tree-rings should not be overemphasized. Dendrochronological evidence has shown that the common signal may be stronger for some sites than for others and therefore there is no possibility of setting an absolute statistical limit for what constitutes correct cross-dating (Wigley et al., 1987). Here we used the threshold for the t-value as 3.5 as this is the most commonly used single value (Baillie and Pilcher, 1973). Nevertheless, the statistical correlations serve as a good guide, but, as already noted by Douglass (1941), the careful visual inspection of the samples should finally guide the decision as to whether a sample is correctly cross-dated. In the case of varves, the visual inspection of samples can also be used to register intra- to inter-varve stratigraphic change. In fact, these geological criteria of non-increment evidence were already used by Sauramo (1918, 1923) and later recommended by De Geer (1940).

An alternative type of statistical cross-correlation analysis of late Weichselian varves was previously carried out in Sweden (Holmquist and Wohlfarth, 1998). Their approach used a logarithmic transformation of varve-thickness diagrams, for which cross-amplitude and phase spectrum were determined, to calculate the correlations at high and low frequencies separately and thus for short and long timescales, respectively. Within the assemblage of visually dated varve data from southeastern Sweden, they concluded that less than 29% and 15% of all analysed connections in their two geographically divided subsets of data were found to display a significant cross-correlation coefficient (Holmquist and Wohlfarth, 1998). While our method of producing statistical cross-dating evidence was considerably simpler, it was found that less than 20% of the diagrams could not be statistically accepted for the chronology. In comparison to the previous statistical analysis (Holmquist and Wohlfarth, 1998), it was not recommended here to use the low-frequency component of the varve-thickness variability as a basis of cross-dating; in addition, the removal of autocorrelation, the construction of mean chronology and its subsequent stripping by removal of the series with less similar varve datings ameliorated the statistical correlations in the present study and could be seen as promising approaches for future studies.

V Conclusions

The incremental cross-dating of annually banded records is constantly gaining ground as a high-precision dating tool in geochronology, and it is anticipated that it will bring even more significant advances over the forthcoming years (Walker, 2005). Cross-dating, using annual increment widths, is a simple method for compound interpretation of visual and statistical extent without a need for highly expensive laboratory facilities. Timescales based on varve-thickness variations have been used for over 100 years to date late Quaternary deposits and map deglacial ice-margin positions. Similarly, dendrochronological techniques have long been utilized for dating the wood materials to calendar years. Considering that both of the disciplines base their dating similarly on the synchrony in the variations of the annual increments, it may be rather surprising that the two disciplines have thus far evolved independently. It could be postulated that the unlinked history may at least partly be a consequence of the differing ‘genotypes’ of the materials. Nevertheless, the ‘phenotypes’, including the varve-thickness diagrams and the measurement series of tree-ring widths, show similarities in their overall statistical characteristics, as demonstrated in the course of this study. In fact, it is these analogies that facilitate the adaptation of dendrochronologically developed correlation approaches into the toolbox of varve science. A cornucopia of advantages was identified when applying these methods to the varve data to assess their temporal connectedness. Moreover, the analyses demonstrated the practices to examine the spatial synchrony of the varve records. These methods were to provide information needed for realistic connections to be made between two or more varve records. Our findings are expected to modernize aspects of Quaternary geochronology in general and the dating approaches of varve records in particular.

Footnotes

Acknowledgements

The useful comments of the two anonymous reviewers are gratefully acknowledged.

The work of SH was supported by the Academy of Finland (122033, 217724). OH thanks the Finnish Graduate School in Geology for funding.

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