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Using more geologically reasonable constraints on radiometric dates from tuffs in a cyclic mixed carbonate-siliciclastic succession in the Donets Basin, Ukraine, newer and more geologically accurate ages of stage boundaries are calculated for the Carboniferous period. These calculations assume astronomically forced (Milankovitch) cyclicity and use radiometric dates of ash layers in the stratigraphic record. Unlike previous work, measurement uncertainties are propagated through calculations to provide mathematically constrained uncertainty in the ages of the boundaries. This work uses more realistic errors in the known dates of zircons in the ash layers and does not assume an a priori value for the Milankovitch long-eccentricity period. These calculations are compatible with other previous methods of determining the boundaries and represent improvement over existing age constraints.

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The stratigraphic record of the Carboniferous as recorded in the Donets Basin, Ukraine, shows regular, repetitive bundling of mixed carbonate-siliciclastic strata cycling from subaerial terrestrial to deeper marine strata and back. This record has been considered an ancient analog to the astronomical forcing (Milankovitch cycles) recorded in detail for the Pleistocene. Recent work by Davydov et al. [2010] suggested refined Carboniferous stage boundaries based upon counting estimations of Milankovitch periodicity in these carbonate sequences and uranium-lead isotope zircon ages for tonsteins located randomly in the sequence. The use of astronomical forcing to increase precision of biologic stage boundaries is a novel approach because unlike most geologic events, which do not have regular periods, orbital properties are generally periodic. In places where periodic events are recorded, they should be used to better constrain geologic dates.

The geology in the Donets Basin has long been interesting for its numerous coal beds. Located in the southeastern part of the Dniepr-Donets Depression on the eastern European craton, the basin is approximately 100–200 km along strike and contains up to 22 km of sediment [Stovba et al., 1996] of nearly continuous deposition during the Carboniferous [Izart et al., 1996; Eros, 2010].

The stratigraphic sequence can be grouped into cyclothems on the order of meters to tens of meters thick that show retrogradational (transgressive) to progradational (regressive) trends inside each cyclothem. These sequences are bounded by evidence of subaerial exposure between the regressive and overlying transgressive phases and can be combined into a framework that may record sea level fluctuations from tens to thousands of kya. Similar sequences are identified in other Pennsylvanian basins [Heckel, 1986; Olszewski and Patzkowsky, 2003; Heckel, 2008; Bishop et al., 2010].

Davydov et al. [2010] used Thermal Ionization Mass Spectrometry (TIMS) to date zircons inside seven tonsteins through the sequence with reported dates from 314.40 ± 0.06 Ma to 328.26 ± 0.11 Ma. One bed, ℓ3, had zircon clusters with two different ages (312.01 ± 0.08 Ma and 312.18 ± 0.07 Ma), but the authors, without explanation, seemed to use the older date so that it would record half a Milankovitch cycle at roughly 200 ka. Although their tonstein zircons had analytic error, the authors did not propagate uncertainties into calculations of their stage boundaries, but instead just argued that the period of intervals was consistent with a roughly 400,000 year Milankovitch cycle and used that to count cycles. They used this value as the period to constrain stage boundaries in the Carboniferous. They furthermore argued that they could constrain their boundaries to approximately 100 ka, the next highest-order Milankovitch period; however, their uncertainty from the two different ages of the same stratigraphic layer (the ℓ3 tonstein-bearing layer), approximately 0.17 ± 0.08 Ma would indicate problems with that approach.

Expanding on the ideas presented by Davydov et al. [2010], but taking a more quantitative approach, the period of cyclicity between any two radiometrically determined layers in a succession characterized by an assumed stratigraphic cyclicity can be calculated. Furthermore, by using more geologically reasonable errors in the eruption dates, in addition to the instrument uncertainty, it is possible to constrain stage boundaries. Applying a more rigorous mathematical procedure, it is possible to successfully propagate uncertainties through calculations and derive error bars on calculated stage boundaries. These methods allow comparisons between calculations using both recorded dates of the ℓ3 ash layer.

3.1. Determining the Milankovitch Period

Using either the stratigraphic column of Izart et al. [1996] or that of Eros [2010] and the uranium-lead dates of Davydov et al. [2010], cyclicity periods are calculated for each combination of known points in the stratigraphy. Each period, P_{ij}, of cyclic events can be calculated as the ratio of the difference in time, T, and the difference in cycle number N between any two (i,j) points in the cycle (conveniently chosen as the dated tonsteins in the column) as calculated in equation (1).

The restriction i < j eliminates both the diagonal elements and the duplicates in the symmetric matrix. The total number of periods, P_{ij} possible with n known points will be (n)(n − 1).

Any error in the knowledge of time should be uncorrelated with any error in the knowledge of cycles, as the time was obtained using radiometric dating techniques on zircons and the cycles were determined using stratigraphic techniques through outcrops and cores. This greatly simplifies the propagation of uncertainty as the covariance will be zero. Also, the error in the time associated with the instrument and the errors in the time due to the unknown amount of time between the formation of the zircon and the eruption are not correlated. One is due to instrument precision and the other due to eruption timing. Therefore, the error in the age of an ash layer, σ_{T}_{i}, can be expressed as σ_{T}_{i}^{2} = σ_{e}^{2} + σ_{i}^{2}, where σ_{e} is the eruption error and σ_{i} is the instrument error for that measurement. The total error in the period can be expressed as in equation (2).

It has been assumed that σ_{e} and σ_{N}do not vary from tonstein to tonstein or through the section. From equation (2), it is clear that the eruptive error is as important as the instrument error in understanding the uncertainty of the Milankovitch period.

3.2. Calculation of Stage Boundaries

While potentially questionable, there is insufficient data to not assume that the Milankovitch period remains relatively constant throughout the 30 million years of interest and will be used as constant over the time period of interest. Using the point-slope form of a linear equation, the age of a stage boundary relative to each of the dated ash layers will be determined. The slope is the Milankovitch period, P_{ij}, the point is the known time and cycle number of a layer k, and the point of interest is the stage boundary, T_{bijk}, for which there is a known cycle number, N_{b}. This is expressed in equation (3).

In order to eliminate all covariance, k is forbidden to be equal to either i or j in equation (3). For ease of notation, each combination of b, i, j, and k will be represented by m. For each calculated stage boundary (with unique combination of b,i,j, and k (or m)), the errors in the calculation are propagated as shown in equation (4).

Carrying these calculations over all possible Milankovitch period and each radiometrically determined time in the Carboniferous, it is possible to calculate average stage boundaries, B, in equation (5).

The uncertainty in this is expressed, in equation (6), as the root-mean-square value of the uncertainties in the individual calculations.

B is the calculated absolute age of a stage boundary, σ_{B} is the associated error of that age, and M represents the total number of combinations, and will be equal to (n) (n − 1) (n − 2), where n is the number of known dates.

4. Numerical Results

For these calculations, cycles were counted from Figure 6 in the study by Davydov et al. [2010] (for the Izart stratigraphic column) and from Appendix IV in the study by Eros [2010] (for the Eros stratigraphic column). Cycle number for the known tonstein ages and the stage boundaries of interest are in Table 1. With seven radiometric dates, there are 21 (nonidentical) Milankovitch periods that can be calculated using the Eros [2010] data for each radiometric date of the bed ℓ3. The Izart et al. [1996] stratigraphic section only contains sequence counting in the Moscovian, and so only six dates can be used in 15 combinations for each radiometric date of the ℓ3 bed. These calculations for periods allow the Eros [2010] column to calculate 105 values for each stage boundary and for each value of ℓ3. The Izart et al. [1996] column will calculate 60 values for each stage boundary for each radiometrically determined date of ash layer ℓ3.

Table 1. Cycle Numbers of Dated Tonalites and Stage Boundaries in the Donets Basin^{a}

Stratigraphic Point

Izart Column

Eros Column

a

The Izart column started at the Moscovian. The ℓ3 layer has zircons dated at both 312.01 ± 0.08 Ma and 312.18 ± 0.07 Ma.

Gzhelian-Permian

30.6

71

Kasimovian-Gzhelian

24.6

64.8

Moscovian-Kasimovian

17.6

56.5

n1 (307.26 ± 0.11 Ma)

16.3

55

m3 (310.55 ± 0.10 Ma)

8

48.5

ℓ3 (see caption)

5.1

43.25

ℓ1 (312.23 ± 0.09 Ma)

4.3

42.5

k7 (313.16 ± 0.08 Ma)

2.8

41.25

k3 (314.4 ± 0.06 Ma)

0.8

38.75

Bashkiran-Moscovian

0

38

Serpukhovian-Bashkiran

not recorded

16.5

c11 (328.26 ± 0.11 Ma)

not recorded

6.25

Visean-Serpukhovian

not recorded

0

Due to the coarseness of the cyclicity in the Izart et al. [1996] column, uncertainty in cycle number, σ_{N}, is taken to be a half cycle. Because of the greater detail in the Eros [2010] column, the uncertainty in cycle number, σ_{N}, was taken at a quarter cycle. The uncertainty in the age of the zircon at the time of eruption is poorly constrained. Because of the two ages from the ℓ3 tonsteins, 312.01 ± 0.08 Ma and 312.18 ± 0.07 Ma, a value greater than 170 ± 70 ka is required. The uncertainty in residence time for the zircon grains, σ_{e}, was taken to be 0 years, 150,000 years, 300,000 years, and 450,000 years for both stratigraphic columns and for both radiometric dates for the ℓ3 ash layer.

The stage boundary results for the Izart et al. [1996] stratigraphic column are summarized in Tables 2 (ℓ3 = 312.01 ± 0.08 Ma) and 3 (ℓ3 = 312.18 ± 0.07 Ma). Comparable results for the Eros [2010] data are summarized in Tables 4 (ℓ3 = 312.01 ± 0.08 Ma) and 5 (ℓ3 = 312.18 ± 0.07 Ma).

Table 2. Stage Boundaries Using Izart et al. [1996] and the Age of ℓ3 Ash Layer as 312.01 ± 0.08 Ma and Average Milankovitch Period With Varying Uncertainties and Using Uncertainty in Cycle Number as One Half

Stage Boundary

Age, Ma

Value of σ_{e}, Ma

0.0

0.15

0.30

0.45

Gzhelian-Permian

299.80 ±

0.45

0.54

0.73

0.98

Kasimovian-Gzhelian

302.73 ±

0.35

0.41

0.56

0.75

Moscovian-Kasimovian

306.14 ±

0.23

0.27

0.37

0.49

Bashkiran-Moscovian

314.73 ±

0.19

0.22

0.29

0.38

Milankovitch Period

0.488 ±

0.037

0.044

0.061

0.081

Table 3. Stage Boundaries Using Izart et al. [1996] and the Age of ℓ3 Ash Layer as 312.18 ± 0.07 Ma and Average Milankovitch Period With Varying Uncertainties and Using Uncertainty in Cycle Number as One Half

Stage Boundary

Age, Ma

Value of σ_{e}, Ma

0.0

0.15

0.30

0.45

Gzhelian-Permian

300.21 ±

0.40

0.50

0.71

0.96

Kasimovian-Gzhelian

303.04 ±

0.31

0.38

0.54

0.73

Moscovian-Kasimovian

306.34 ±

0.21

0.25

0.35

0.48

Bashkiran-Moscovian

314.63 ±

0.17

0.20

0.28

0.37

Milankovitch Period

0.471 ±

0.033

0.041

0.058

0.079

Table 4. Stage Boundaries Using Eros [2010] and the Age of ℓ3 Ash Layer as 312.01 ± 0.08 Ma and Average Milankovitch Period With Varying Uncertainties and Using Uncertainty in Cycle Number as One Quarter

Stage Boundary

Age, Ma

Value of σ_{e}, Ma

0.0

0.15

0.30

0.45

Gzhelian-Permian

299.55 ±

0.36

0.47

0.72

1.00

Kasimovian-Gzhelian

302.38 ±

0.30

0.41

0.61

0.86

Moscovian-Kasimovian

306.17 ±

0.24

0.32

0.49

0.68

Bashkiran-Moscovian

314.61 ±

0.16

0.22

0.33

0.46

Serpukhovian-Bashkiran

324.42 ±

0.27

0.35

0.54

0.75

Visean-Serpukhovian

331.96 ±

0.40

0.54

0.81

1.13

Milankovitch Period

0.454 ±

0.022

0.029

0.044

0.061

Table 5. Stage Boundaries Using Eros [2010] and the Age of ℓ3 Ash Layer as 312.18 ± 0.07 Ma and Average Milankovitch Period With Varying Uncertainties and Using Uncertainty in Cycle Number as One Quarter^{a}

Stage Boundary

Age, Ma

Value of σ_{e}, Ma

0.0

0.15

0.30

0.45

a

Table 5 shows the best calculated stage boundaries with the boldface representing the best uncertainties.

Gzhelian-Permian

300.15 ±

0.33

0.46

0.71

1.00

Kasimovian-Gzhelian

302.87 ±

0.28

0.39

0.60

0.85

Moscovian-Kasimovian

306.51 ±

0.23

0.31

0.48

0.67

Bashkiran-Moscovian

314.63 ±

0.15

0.21

0.32

0.46

Serpukhovian-Bashkiran

324.07 ±

0.25

0.34

0.53

0.74

Visean-Serpukhovian

331.31 ±

0.38

0.52

0.80

1.12

Milankovitch Period

0.439 ±

0.020

0.027

0.043

0.060

The Milankovitch periods for the Izart et al. [1996] column using the younger (older) tonstein and an average value of 488 (471) ka. The Milankovitch periods for the Eros [2010] column using the younger (older) tonstein had an average value of 454 (439) ka. The calculated periods are slightly higher than the modern values of 413,000 years [Fischer, 1986].

The Milankovitch periods for the Izart et al. [1996] column using the younger tonstein (ℓ3 = 312.01 ± 0.08 Ma) ranged from 275 ± 602 ka to 620 ± 415 ka, with an average value of 488 ka. The Milankovitch periods for the Izart et al. [1996] column using the older tonstein (ℓ3 = 312.18 ± 0.07 Ma) ranged from 63 ± 552 ka to 620 ± 415 ka, with an average value of 471 ka. The Milankovitch periods for the Eros [2010] column using the younger tonstein (ℓ3 = 312.01 ± 0.08 Ma) ranged from 278 ± 86 ka to 930 ± 550 ka, with an average value of 454 ka. The Milankovitch periods for the Eros [2010] column using the older tonstein (ℓ3 = 312.18 ± 0.07 Ma) ranged from 67 ± 587 ka to 930 ± 550 ka, with an average value of 439 ka. The calculated periods are slightly higher than the modern values of 413,000 years [Fischer, 1986].

With equation (2), it is apparent that the uncertainty decreases as the distance between two measured points increases (both in the stratigraphic column, N, and the ages, T) so the lowest error should be at combinations of i and j such that these distances are greatest. Many of these separated calculations show a period between 395,000 years and 440,000 years in the Eros [2010] data. Because the cyclicity data based on Izart et al. [1996] is not as complete in the older record, the periods are higher, between 440,000 years and 540,000. Only the data in the Eros [2010] data allows period calculations for ΔN > 10 cycles, and in these, the Milankovitch period only ranges from 419 ka to 442 ka.

Because the stratigraphic section in the study by Eros [2010] is more detailed and covers cyclic behavior for a longer period of time, the results based upon the Eros [2010] stratigraphic column is preferred. Because the youngest zircon is a better choice for constraining eruptions in the ℓ3 tonstein, the results in Table 4 are preferred to Table 5. Furthermore, because an uncertainty of 300,000 years for residence time for the zircons is consistent with the spread of zircons in the ℓ3 tonsteins and other work in volcanic systems [Brown and Fletcher, 1999; Cooper and Reid, 2008], the third column of uncertainties in Table 4 are reported as the stage boundaries. This selection is also most consistent with work in the Urals that found the Carboniferous-Permian boundary at 298.90_{−0.15}^{+0.31} Ma [Ramezani et al., 2007].

Compared to the stage boundaries earlier published [Davydov et al., 2004; Menning et al., 2006; Davydov et al., 2010], this analysis shows generally good agreement with previous results shown in Table 6. The Bashkirian and Serpukhovian are calculated as starting slightly earlier than Davydov et al. [2010] found in their analysis. The uncertainties in these calculations are much improved over the results published in the Geologic Time Scale 2001 [Davydov et al., 2004]. The uncertainties in the study by Davydov et al. [2010] are discussed in section 5.

Table 6. Comparison of Carboniferous Stage Boundaries From Prior Works^{a}

This mathematical treatment of cyclostratigraphy in the Carboniferous is very robust; each stage boundary is determined via 105 individual calculations. Allowing k = i and k = j in the stage boundary equation (equation (3)) changes boundaries in the 10,000 year place. Those calculations involve 147 calculations as increased robustness. The Geologic Time Scale 2004 (GTS2004) stage boundaries are presented with error bars [Davydov et al., 2004], although it is not clear how these bars were derived. They seem to be increasing over time, but it is not clear how these increases were determined. Davydov et al. [2010] do not include error bars in their results, but explain in text that biostratigraphic boundaries have been tuned with “a high-resolution (∼100 ka) calibration.” That assertion is inconsistent with their data, as the ℓ3 tonstein demonstrates that any one zircon set in the same stratigraphic location can have an age at least 170,000 years from the youngest eruption date. Even ignoring the eruption ages and uncertainties in cycle number, these equations show that the uncertainty in stage boundaries just from instrument error ranges from 0.10 to 0.18 Ma. The presently presented stage boundaries have clearly defined uncertainties that account for the errors in the instrument, the uncertainties in zircon residence time in a magma chamber, and errors in cycle counting. They represent much more reasonable uncertainties in stage boundaries from 300 million years ago and are much improved over previous methods.

Also, from this analysis, the start of the Moscovian, Bashkirian, and Serpukhovian are all pushed backward with respect to the GTS2004. The most recent calculations [Davydov et al., 2010] have values more similar to the values presented here for the Moscovian, but their values for the Bashkirian and Serpukhovian are younger than calculated here. This lack of agreement is potentially due to utilization of a ∼400 ka “calculated cycle duration” [Davydov et al., 2010]. The work presented here has shown that the period is not 400 ka, but higher. This would move those stage boundaries to older times. Also, their need to modestly reinterpret the upper Gzhelian in order to reach the Carboniferous-Permian boundary would also be lessened by a longer Milankovitch long-eccentricity period.

6. Conclusion

Because the Eros [2010] hierarchical stratigraphy used objective criteria for defining genetic depositional units and covers the longest time period, the results in Table 4 are suggested as the best stage boundaries calculated by this method using the second column of uncertainties. This set of calculations also aligns the end of the Carboniferous most closely with work published by Ramezani et al. [2007].

Detailed stratigraphy has been recorded in the Donets Basin [Eros, 2010] and smaller cycles are often identified within these major cycles. These higher-frequency cycles may be related to additional short astronomical parameters. The period of the most obvious higher-order cycle, which appears to be around one-quarter of the roughly 440–450 ka period, has potential in greater refinements of the geologic time scale, but more detailed work must be done before it can be used for tuning the stratigraphic record.

Notation

i,j,k,m

Arbitrary indicies.

P_{ij}

Period of cyclicity between radiometric times i and j, in Ma.

N_{i}

Cycle number at a given time, dimensionless.

T_{i}

Time measured from zircon ages, in Ma.

T_{bijk}

Time of boundary b, using starting date and cycle k and period P_{ij}, in Ma.

T_{m}

Identical to T_{bijk}, in Ma.

σ_{P}

Uncertainty of the period of cyclicity, in Ma.

σ_{Ti}

Uncertainty of a radiometric time, in Ma.

σ_{e}

Uncertainty of the residence time of zircon in magma, in Ma.

σ_{i}

Uncertainty in measurement of zircon age, in Ma.

σ_{N}

Uncertainty from counting cycles, dimensionless.

σ_{Tm}

Uncertainty of an individual stage boundary calculation, in Ma.

σ_{B}

Uncertainty of the final age of the stage boundary, in Ma.

B

Age of stage boundary, in Ma.

M

Number of independent combinations for averaging, dimensionless.

Acknowledgments

I want to thank Isabel P. Montañez for helpful discussions previous to and during my work on this project. Mike Eros and Lauren G. Martin also provided insight on the stratigraphic record in the Carboniferous. Gary Eppich provided insight on uranium istotope systematics. Philip H. Heckel provided a very detailed review, which helped to clarify the writing. An anonymous reviewer also provided comments that helped improve the paper.