1 Introduction

The purpose of the geologic time scale (GTS) is to calibrate the long history of our planet Earth in linear time. It guides our understanding, with a focus on the succession of physical, geochemical and biological events in the global sedimentary rock record. This calibration to linear (deep) time is based on joining the chronostratigraphic scale, which is based on observations at the surface of the Earth, with chronometric measurements (in millions of years) made on small rock samples.

For clarity and precision in international communication, the global rock record of Earth’s history is subdivided into a “chronostratigraphic” scale of standardized global stratigraphic units, such as “Devonian,” “Miocene,” “Zigzagiceras zigzag ammonite zone,” or “polarity Chron M18r.” Unlike the continuous ticking clock of the “chronometric” scale (measured in years before the year 2000 AD), the chronostratigraphic scale is based on relative time units in which global reference points at boundary stratotypes define the limits of the main formalized units, such as “Cretaceous.” The chronostratigraphic scale is an agreed convention, whereas its calibration to linear time is based on discovery and estimation (Fig. 1).

Fig. 1
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(Source: Gradstein et al. 2020, Fig. 1.1)

The construction of a geologic time scale is the merger of a chronometric scale (measured in years) and a chronostratigraphic scale with formalized definitions of geologic stages, biostratigraphic zonation units, magnetic polarity zones and other subdivisions of the rock record

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In contrast to the Phanerozoic (538 Ma to Recent), Precambrian stratigraphy (approximately 4,500 to 538 Ma) is formally classified chronometrically; that is, the base of each Precambrian eon, era and period is assigned a numerical age (Table 1). We refrain from dealing with Precambrian. The new geologic time scale as presented in GTS2020 (Gradstein et al. 2020) is shown in Fig. 2. The nearly 100 stages in the Phanerozoic vary in duration between 1.5 and more than 20 million years, as a function of consensus definitions using classical lithostratigraphy, biota evolution and other major types of events in deep time. In the next section, attention is briefly given to the philosophy and actual data contributing to geologic time scale building and its linear scale.

Table 1 Current framework for subdividing Earth stratigraphy
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Fig. 2
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Geologic Time Scale 2020

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2 Geologic Time Scale GTS2020

2.1 Methods and Ages

Figure 3 schematically shows methods used for the construction of Geologic Time Scale 2020 (GTS2020). Supersplining as applied in this paper was originally applied to the Paleozoic interval only (Gradstein and Agterberg 2022). In this, more recent, study it is applied to the combined Paleozoic and Mesozoic intervals. For the Triassic through middle Jurassic time interval, stratigraphic reasoning has been a prime method in GTS2020 and earlier scales, which hampers determination of proper error bars. For this reason, the dates for this interval are newly included in the superspline for which other inputs are the same as those used as inputs the for spline curves fitted for separate periods or combinations of periods for GTS2020 (Gradstein et al. 2020).

Fig. 3
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(Source: Gradstein et al. 2020, Fig. 1.3)

Methods used to construct geologic time scale GTS2020 integrate different techniques depending on the quality of data available within different intervals

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In Table 2 the ages of stage boundaries in GTS2020 are compared with those in GTS2012. Differences in age of 0.5 million years or more include new dates for the early Permian, parts of Devonian and Ordovician.

Table 2 Modified ages of stage boundaries in Geologic Time Scale 2020 (GTS2020) relative to Geologic Time Scale 2012 (GTS2012)
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2.2 Combining Dates at the Same Stratigraphic Level

Two or more samples on which the age was determined may occur at the same stratigraphic level. Such different age determinations should then be averaged, and the increase in the precision of the resulting average age should also be calculated. The method described below for two ages at the same stratigraphic level is readily generalized when there are more than two age determinations. It can also be used to combine estimates for the same epoch or period boundary obtained by two different methods, as discussed in more detail elsewhere (Agterberg et al. 2020). Suppose that yi and yi+1 are two successive age determinations with the same x-value. Their approximate 95% confidence interval can be written as 2.σ(yi) and 2. (yi+1). From these two values, the weights of the two observations can be computed as follows: The sum of the weights wi = 1/σ2(yi) and wi+1 = 1/σ.2(yi+1) can then be written as w(x) = wi + wi+1where x represents location of both yi and yi+1 along the x-axis. If there are two dates only, their weighted average is

$$ y\left( x \right) = \frac{{w_{i} y_{1} + w_{i + 1} y_{i + 1} }}{w\left( x \right)}. $$

This method was also used for combining two different estimates at a series boundary. Obviously, the procedure is readily extended when there are more than two age determinations.

2.3 Spline Fitting

Various spline-based methods have been used in the past for estimating the numerical geologic time scale (GTS). Comprehensive reviews were given in the geomathematical method chapters of GTS2004 (Agterberg et al. 2004), GTS2012 (Agterberg et al. 2012) and GTS2020 (Agterberg et al. 2020). For Paleozoic periods in the earlier (pre-2020) GTS publications, time scale construction included fitting a cubic smoothing spline curve. Results for the version of this technique used for GTS2020 are shown in Figs. 4, 5, 6, 7 and 8, in which the data points are more or less evenly distributed along the x-axis. Other statistical techniques, including LOWESS and multiple polynomial regression, produce best-fitting curves that are close to the cubic spline solutions. For a detailed comparison of the cubic spline solution of Fig. 4 with LOWESS and polynomial regression results, see Goldman et al. (2020).

Fig. 4
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(Source: Goldman et al. 2020, Fig. 20.15C)

Best fitting spline curve for the composite of the Ordovician–Silurian graptolite ranges and events

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Fig. 5
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(Source: Becker et al. 2020, Fig. 22.14)

Two-way graphics plot of the scaled zonation in Gradstein et al. (2020) versus radioisotopic dates for construction of the Devonian time scale, using cubic splining and error analysis. Note that the Silurian–Devonian boundary age in GTS2020 is based on the more detailed and updated Ordovician–Silurian spline of Fig. 4

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Fig. 6
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(Source: Aretz et al. 2020, Fig. 23.8)

Best-fitting spline curve for the Carboniferous–Permian

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Fig. 7
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(Source: Gale et al. 2020, Fig. 27.10)

Early Cretaceous and Late Jurassic spline fit and geologic time scale. Spline-fit errors are small, if present, and may be invisible. For details see Agterberg et al. (2020, Ch. 14A: Geomathematics and statistical procedures

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Fig. 8
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(Source: Gale et al. 2020, Fig. 27.11)

Late Cretaceous spline fit and the Late Cretaceous time scale. Spline-fit uncertainties are small, if present, and may be invisible. For details see text in Gale et al. (2020); and in Agterberg et al. (2020, Ch. 14A: Geomathematics and statistical procedures)

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The error bars shown in Figs. 4, 5, 6 and 7 are based on ±2-sigma for the age determinations and widths of rectangular uncertainty boxes for the stratigraphic positions. Only about 5% of these crosses should not intersect the spline curve as expected. However, this condition is not fulfilled for the Late Cretaceous spline (Fig. 8), in which the error bars on the age dates are too narrow (cf. Agterberg et al. 2020, Section 14A.5.8).

Unless the stratigraphic uncertainty can be neglected in the spline fitting, each value is assumed to have a total variance that is the sum of variances for stratigraphic uncertainty with rectangular error boxes of width q and s(x) = 1.15·q/4 (cf. Agterberg, 2004). A cubic smoothing spline f(x) is fully determined by n pairs of values (xi, yi), the standard deviations of the dates s(yi) and a smoothing factor (SF) representing the square root of the average value of the squares of scaled residuals ri = (yi – f(xi))/s(yi). As explained in more detail in Agterberg (2004), the “leave-one-out” cross-validation (CV) method can be used to determine the optimum smoothing factor. In this method, all observed dates yi, between the oldest and youngest one, are successively left out from spline fitting with preselected trial values of SF. The result is (n – 2) spline curves for each SF tried.

Our GTS2020 applications made use of the R program smooth-spline developed by B.D. Ripley and M. Maechler in the R stats package version 3.6.0. Best-fitting smooth curves were obtained by CV in all GTS2020 applications, considering the relative abundance of dates over time in the spline fitting, in addition to consideration of the age determination errors and the stratigraphic uncertainty.

2.4 95% Confidence Intervals on the Spline Curves

Paleozoic time scale construction followed in GTS2004 was based on the idea that a plot of the observed age determinations against the estimated spline ages is approximately according to a straight line, with the simple equation y = x. This procedure is equivalent to the method originally used by McKerrow et al. (1985). Using Student’s t and the standard error se, the 95% confidence belt for a best-fitting straight line at any point (xk, yk) satisfies \({y}_{k}\pm t(n-2)\cdot {s}_{e}\sqrt{1-{R}_{k}},\) where \({R}_{k}=\frac{1}{n}+\frac{{{\{x}_{k}-(\sum {x}_{k}/n)\}}^{2}}{\sum ({{(x}_{k}-(\sum {x}_{k}/n))}^{2}}\) (cf. Agterberg 1974, Eq. 8.31).

For the Ordovician–Silurian spline of Fig. 4, this 95% confidence interval is shown in Fig. 9. Clearly, precision decreases in both time directions away from the center of the best-fitting spline curve.

Fig. 9
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(Source: Agterberg et al. 2020, Fig. 14A.8)

Two-sigma values for estimated spline curve for Ordovician–Silurian graptolites shown in Fig. 4

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Figure 10 shows the 95% half-confidence belt for the Devonian as a solid line. For this period only, the approach in GTS2020 was taken one step further to account for the probable variations in density of sampling points (Agterberg er al., 2020).

Fig. 10
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(Source: Agterberg et al. 2020, Fig. 14A.11)

Approximate uncertainty factor (broken line) used to widen 95% confidence belt (solid line) incorporating changes in density of dates along the Devonian time scale. See text for further explanation

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3 Superspline Fitting

From the standpoint of practicality, the 2004 and 2012 versions of the geologic time scale applied separate spline fittings for Ordovician–Silurian, Devonian and Carboniferous–Permian. To this splined trio were added the Cretaceous and Late Jurassic in GTS2020. With individual chapter authors generating their unique composite standard or other dedicated spline input data per period, it was not really feasible to splice individual period splines in a stratigraphically longer superspline or other concatenated ages solution prior to publication of the individual time scale books.

From the standpoints of geomathematics and stratigraphy it may be argued that a stratigraphic and numerical longer solution, not artificially truncated at period boundaries, would be desirable. Successions of detailed and precise age dates on each site of period boundary (above it and below it) interact with each other. Hence, with the completion of the detailed and dedicated data for GTS2020, the opportunity was created to consider bridging the stratigraphic boundaries of consecutive periods and build a long superspline.

For the Paleozoic in GTS2020, a superspline was already fitted (see Fig. 11) and its predicted stage boundaries compared with those in GTS2020 (Gradstein and Agterberg 2022). In order to combine the separate splines for different periods or combinations of periods, their different chronostratigraphic scales were linearly rescaled to obtain a single scale for the entire Paleozoic. These linear transformations of chronostratigraphic scales were carried out on the basis of “anchor ages” coinciding with period boundaries (cf. Table 3). For this paper the same procedure was followed for the Mesozoic.

Fig. 11
figure 11

(Source: Gradstein and Agterberg 2022)

Paleozoic superspline. The unit for the chronostratigraphic scale runs from 0 for the base of the Ordovician to 1 for the top of the Permian. Within each of the three intervals between anchor points the chronostratigraphic scale values are linearly related to those used to estimate the GTS2020 stage boundary ages

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Table 3 Generally small or negligible differences in age of stage boundaries for GTS2020 and the superspline method
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In total, nine anchor ages were selected to allow the eight linear transformations required to obtain a single Paleozoic–Mesozoic chronostratigraphy. Seven of the nine anchor ages were set at epoch boundaries including the Permian–Triassic and the Cretaceous–Paleogene boundaries which are period boundaries as well. Estimated superspline values at the Permian–Triassic Epoch boundary are both equal to the input values set at 251.9 Ma. A potential drawback of the supersplining results with respect to the GST2020 results, most of which were obtained by fitting separate splines, is that input uncertainties at all nine anchor ages were set equal to zero in order to allow for the required linear transformations of the GTS2020 chronographic scales.

For the Indian-Callovian we used a bio-chronostratigraphic stratigraphic scale estimated from the zonal range assignments and biostratigraphy listed in the Appendix 2 by Schmitz (2020) on the radioisotopic ages used in GTS2020. Again, each data set with modified chronostratigraphic scale xn has two anchor points set at the beginning and end of each period. In Fig. 12 the value for base of the Triassic was set equal to 1 and that for the top of the Cretaceous equal to 2. The estimated Mesozoic superspline ages are shown in Table 3. It remains an open question whether the superspline values for the Paleozoic and Mesozoic of Table 3 are slightly better than the GTS2020 estimates. The 0.9 Ma difference in age of Late Cretaceous stage boundaries is thought to be due to underestimation of uncertainties in the vital data, and sizable −2.0 through 2.3 Ma age differences in Aalenian through Kimmeridgian stage boundaries indicate insufficient accurate and precise vital stratigraphic information. Both Rhaetian and Norian stages lack accurate and precise age and zonal information, and remain poorly defined in terms of the geologic time scale. No ready explanation may be given for the discrepancies in age between GTS2020 and the superspline ages for Olenekian and Hirnantian, but otherwise it may be noted that the Paleozoic composite standards and constrained optimization fits and spline fits provided an excellent basis for the current superspline with negligibly small numeric deviations.

Fig. 12
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Mesozoic superspline. The unit for the chronostratigraphic scale runs from 1 for the top of the base of the Triassic to 2 for the top of the Cretaceous. See Table 3 for the two other anchor points

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4 Conclusions

Modern geologic time scale building includes scaling composite zonal standards and tuning Milankovitch cycle segments. As a function of the situation that a majority of stratigraphers specialize on selected periods and stages, GTS2020 emphasizes building the linear scale period by period, while care is taken to provide reliable overlap and connection.

It may be argued that a stratigraphic and numerical longer solution, not artificially truncated at period boundaries, would be desirable. Ideally, there should be a single straight-line relationship between age and chronostratigraphy. A disadvantage of the current superspline is that its anchor points are assumed to be error-free. It remains an open question whether the superspline values for the Paleozoic and Mesozoic of Table 3 are slightly better than the GTS2020 estimates. The 0.9 Ma difference in age of Late Cretaceous stage boundaries is thought to be due to underestimation of uncertainties in the vital data, and sizable −2.0 through 2.3 Ma age differences in Aalenian through Kimmeridgian stage boundaries indicate insufficient accurate and precise vital stratigraphic information. Both Rhaetian and Norian stages lack accurate and precise age and zonal information, and remain poorly defined in terms of the geologic time scale. No ready explanation may be given for the discrepancies in age between GTS2020 and the superspline ages for Olenekian and Hirnantian, but otherwise it may be noted that the Paleozoic composite standards and constrained optimization fits and spline fits provided an excellent basis for the current superspline with negligibly small numeric deviations.