Las Funciones Gerenciales.

Once crossdating had been carried out on the raw data, a site mean chronology was constructed. This procedure has multiple steps (Cook and Briffa, 1990). Firstly, the GI measurements were detrended to remove the ontogenetic growth trend present in the individual series (Figure 4.4). Detrending fits a series specific data adaptive function to the raw data using ordinary least squares and then removes the trend of this function using either division or subtraction. The result of this is a dimensionless index time-series with no age- related/ontogenetic trend. For this study the main method of detrending used was a negative exponential (NE) function (see Equation 4.1), or where this did not fit a linear function was used, using programme ARSTAN (Cook, 1985b). Division rather than subtraction was chosen as

Figure 4.3: Two Standardised Growth Indexes (SGI) detrended series for shell C1-L2, one from the outer shell (Daniels, 2010 –red) and the other from the tooth (Stott et al., 2010 – black), the two series share a high visual coherence. The corresponding r and p-values between the two series (0.77 and 0.000 respectively) indicate that there is a common signal between the two SGI series.

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this helps stabilise the variance of the final series between the juvenile and more mature phases (Cook and Peters, 1997). For some series, detrending using the NE or linear function was not suitable as the fitted function went below zero resulting in indices greater than infinity. In these few situations a Hugershoff function (see Equation 4.2) was used (see Figure 4.4). A Hugershoff function was used as it is more flexible than the NE function and still retains some potential lower frequency information recorded in the growth records; as a result it is less likely to go below zero therefore minimising end effect index inflation. However, for very short series, as those seen in site C6, NE or Hugershoff functions are not appropriate (Figure 4.5) as the functions are essentially too ‘stiff’ for the short record and would go below zero. In these cases, a 10 year smoothing spline (Figure 4.5b) was used for detrending with the caveat that such a flexible option would remove any potential climatic information at time-scale longer than 5 years.

Where:

Gt is the growth trend of the raw data

a is the growth intercept of the function at year t e is the exponential function

b is the decay constant

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Figure 4.4: A) Graph illustrating the ontogenetic growth trend in the raw growth in A. islandica GIs and a typical negative exponential function detrending curve fitted to the data to remove this trend for shell C8-L3. B) Application of Hugershoff detrending to another A. islandica GI series. C and D illustrate the detrended SGI for each of the shells for shell C8-L5. C) Illustrates the detrended shell C8L3 series and D) shows the

detrended series for shell C8L5.

Figure 4.5: Examples of the application of A) a linear detrending function and B) a Hugershoff detrending function to the raw growth series for shell C6-L5 as seen in ARSTAN output when the functions go below zero and are therefore inappropriate for detrending.

A

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The use of a NE function to detrend data is not new within sclerochronological literature. Strom et al. (2004), Strom et al. (2005) and Nielsen et al. (2008) used this method, but with different mollusc species, while Butler et al. (2009a; 2010) and Stott et al. (2010) applied NE function detrending to A. islandica. Witbaard et al. (2003) used a NE function to detrend A. islandica growth records, but this was in conjunction with a 66-year spline as a second processing step. Such a practice is also carried out by some dendrochronologists as it is believed that such a double detrending approach removes the juvenile growth and then secondly reduces the residual noise present within the record (Borgaonkar et al., 1999). However, such an approach also removes potential climatically driven long-term variability in the series. As the NE function both removes the ontogenetic growth trend and preserves the longer-term variability in the shell at frequencies up to the mean length of the samples (Cook et al., 1995), a double detrending approach is not used in this study.

Once the raw data have been detrended, the resulting individual index series are averaged together to derive a site specific mean index master chronology. The robustness of the mean chronology is related to both the number of series used and the strength of the common signal between them (measured by RBAR, where RBAR is defined here as the inter-series correlation between all possible pairs of time-series in the sample). The weaker the common signal, the greater the number of series needed to derive a robust mean chronology. It is common practice to use signal strength statistics in dendrochronology to assess the ‘quality’ of the resultant chronology. The seminal paper describing relevant signal strength statistics is that by Wigley et al. (1984) where the Expressed Population Signal (EPS) is derived. Essentially, the EPS can be thought of as an empirical assessment of how the average of a sample of time-series correlates with the theoretical infinitely replicated population time-series. The derivation of the EPS can be described as dividing the signal by the total variance (signal + noise). The EPS is calculated using the following equation:

Where EPS is the Expressed Population Statistic value n is the number of time series

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Wigley et al. (1984) showed that an EPS value greater than 0.85 was desirable to ensure a robust mean chronology. The EPS equation (Equation 4.3) can be rearranged (see Wilson and Elling, 2004 for an applied example) to determine how many shell series would be needed to derive a robust chronology:

̂

Where EPS(x) is the 0.85 value suggested by Wigley et al. (1984) – although other values can be used

̂ is the predicted number of series required to produce a robust chronology

̅ is the inter-series correlation between the series in the master chronology.

As well as calculating the EPS, ̂ and ̅ values for all of the chronologies after they were detrended, the EPS value was also calculated once the series had been transformed using first differencing (FD) to provide a robust assessment of the inter-annual signal.

For the master chronology of each site, the EPS statistics for both the whole chronology (WC) period and period of maximum replication (PMR), i.e. the time frame for which all shells in a chronology are present, are detailed in Section 4.3.1. The RBAR for these periods are calculated and then these data are used to work out the theoretical number of shells required to reach an EPS value of 0.85 (n value as shown in Equation 4.3). Although not ideal, those chronologies where the EPS is below the required 0.85 value will still be compared to the instrumental datasets in Section 4.3.4 with the caveat that any results are purely preliminary and intended to be used only as a preliminary guide of how shells from different sites may be responding to different climatic and environmental conditions.

4.2.4.1Inter site comparisons

If growth is dominated by regional influences (e.g. climate), then there should be a degree of common variability between the site chronologies. To determine whether there is a common signal between the sites, the chronologies (both unfiltered and FD) were compared using

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correlation analyses. The correlations were carried out with ±3 year lags to look for potential leads/lags between the series. While the outermost GI anchors the shell chronology to the time of ‘live’ collection, there can be difficulties in counting the outermost few GIs (see Section 4.2.2). This is because when creating a peel, the most recent GIs sometimes do not appear clearly on the image due to an edge effect where the shell and resin block meet. In addition, damage to the shell can cause the removal of GIs (although this is less common in the umbo compared to the outer shell).