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Analyzing Statistical Age Models to Determine the Equivalent Dose and Burial Age Using a Markov Chain Monte Carlo Method

Jun Peng
Jun Peng
   | 31 dic 2021
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Pubblicato online: 31 dic 2021
Pagine: 147 - 160
Ricevuto: 07 gen 2019
Accettato: 12 lug 2019
DOI: https://doi.org/10.1515/geochr-2015-0114
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© 2019 Jun Peng, published by Sciendo
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Fig. 1

Statements used to conduct MCMC sampling according to the CAM and the MAM3 (log-scale) using De sets from individual samples. The MXAM3 can be applied in a similar manner to the MAM3 by changing the likelihood function.
Statements used to conduct MCMC sampling according to the CAM and the MAM3 (log-scale) using De sets from individual samples. The MXAM3 can be applied in a similar manner to the MAM3 by changing the likelihood function.

Fig. 2

Statements used for conducting MCMC simulation simultaneously according to different age models (log-scale) using De sets from three samples whose ages were constrained such that a1<a2<a3. The three samples were analyzed using the MAM3, CAM, and MXAM3, respectively. The protocol can be easily adapted for MCMC sampling without order constrains for burial ages.
Statements used for conducting MCMC simulation simultaneously according to different age models (log-scale) using De sets from three samples whose ages were constrained such that a1<a2<a3. The three samples were analyzed using the MAM3, CAM, and MXAM3, respectively. The protocol can be easily adapted for MCMC sampling without order constrains for burial ages.

Fig. 3

Measured multi-grain De distributions for four aeolian samples from Gulang County at the southern margin of the Tengger Desert, China (Peng et al., 2016b). N denotes the number of measured aliquots. Note that in Peng et al. (2016b) an additional uncertainty of σb=5% was added in the quadrature to the RSEs of the measured De values to account for sources of errors that were not considered in the measurements. However, no additional uncertainty was added to the De distributions shown here.
Measured multi-grain De distributions for four aeolian samples from Gulang County at the southern margin of the Tengger Desert, China (Peng et al., 2016b). N denotes the number of measured aliquots. Note that in Peng et al. (2016b) an additional uncertainty of σb=5% was added in the quadrature to the RSEs of the measured De values to account for sources of errors that were not considered in the measurements. However, no additional uncertainty was added to the De distributions shown here.

Fig. 4

Posterior distributions of burial ages obtained from the CAM for the four aeolian samples taken from the same sedimentary section obtained using the MCMC sampling protocol, as shown in Fig. 2. The blue-coloured density plots denote results obtained by constraining the burial ages in ascending order. The grey-coloured density plots denote results obtained by imposing no constraints on the order of burial ages. The value inside parentheses denotes the calculated RSD of burial age.
Posterior distributions of burial ages obtained from the CAM for the four aeolian samples taken from the same sedimentary section obtained using the MCMC sampling protocol, as shown in Fig. 2. The blue-coloured density plots denote results obtained by constraining the burial ages in ascending order. The grey-coloured density plots denote results obtained by imposing no constraints on the order of burial ages. The value inside parentheses denotes the calculated RSD of burial age.

Fig. 5

Distribution of OSL sensitivity for the 127 grains of a sample taken from Lake Mungo, Australia. The red line denotes the probability density curve obtained by fitting the OSL sensitivity data using a Gamma distribution. The fitted Gamma distribution has a shape parameter of α=0.565 and a rate parameter of β=0.191. Q[x%] denotes the x% sample quantile of OSL sensitivity.
Distribution of OSL sensitivity for the 127 grains of a sample taken from Lake Mungo, Australia. The red line denotes the probability density curve obtained by fitting the OSL sensitivity data using a Gamma distribution. The fitted Gamma distribution has a shape parameter of α=0.565 and a rate parameter of β=0.191. Q[x%] denotes the x% sample quantile of OSL sensitivity.

Fig. 6

Simulated De distributions of nine samples from a sedimentary sequence. Each De set contains 100 De values (and associated standard errors). The shaded bars in each plot indicate the burial dose within 2σ error. The first three samples (#1–#3) are partially-bleached, the next three samples (#4–#6) are fully-bleached, and the last three samples (#7–#9) contain fully-bleached grains that have been subsequently mixed with younger, intrusive grains. The nine samples are assumed to be collected from the same section but at different depths, and their estimated burial dose and age are summarized in Fig. 7 and Table 3.
Simulated De distributions of nine samples from a sedimentary sequence. Each De set contains 100 De values (and associated standard errors). The shaded bars in each plot indicate the burial dose within 2σ error. The first three samples (#1–#3) are partially-bleached, the next three samples (#4–#6) are fully-bleached, and the last three samples (#7–#9) contain fully-bleached grains that have been subsequently mixed with younger, intrusive grains. The nine samples are assumed to be collected from the same section but at different depths, and their estimated burial dose and age are summarized in Fig. 7 and Table 3.

Fig. 7

Posterior distributions of burial ages of the nine simulated samples from a sedimentary section obtained using the MCMC sampling protocol, as shown in Fig. 2. Samples #1–#3, #4–#6, and #7–#9 were analyzed using the MAM3, CAM, and MXAM3, respectively. The blue-coloured density plots are results obtained by constraining burial ages in ascending order. The grey-coloured density plots are results obtained by imposing no constraints on the order of burial ages. The first and second values inside parentheses denote the calculated RSD and RE of the burial age, respectively.
Posterior distributions of burial ages of the nine simulated samples from a sedimentary section obtained using the MCMC sampling protocol, as shown in Fig. 2. Samples #1–#3, #4–#6, and #7–#9 were analyzed using the MAM3, CAM, and MXAM3, respectively. The blue-coloured density plots are results obtained by constraining burial ages in ascending order. The grey-coloured density plots are results obtained by imposing no constraints on the order of burial ages. The first and second values inside parentheses denote the calculated RSD and RE of the burial age, respectively.

Fig. 8

Convergence diagnostics for posterior samples of the CAM burial age from sample GL2-1. The plots were automatically generated using function mcmcSAM(). (A) denotes the trace plot; (B) denotes the density plot; (C) denotes the autocorrelations plot, and (D) denotes the Gelman-Rubin convergence diagnostic plot generated using three parallel Markov chains.
Convergence diagnostics for posterior samples of the CAM burial age from sample GL2-1. The plots were automatically generated using function mcmcSAM(). (A) denotes the trace plot; (B) denotes the density plot; (C) denotes the autocorrelations plot, and (D) denotes the Gelman-Rubin convergence diagnostic plot generated using three parallel Markov chains.

A summary of burial dose and age estimated from various statistical age models for nine simulated De sets taken from the same sedimentary sequence. Results obtained using MLE and results obtained using the MCMC sampling protocol as shown in Fig. 2 without and with order constraints are presented. The systematic error on the dose rate measurement shared by all simulated samples was assumed to be σdc=0.

Sample Model Age (ka) Burial dose (Gy) MLE MCMC (without constraints) MCMC (with constraints)
μ (Gy) a (ka) μ (Gy) a (ka) μ (Gy) a (ka)
#1 MAM3 7 18 18.35 ± 1.17 7.95 ± 0.75 17.85 ± 1.17 7.82 ± 0.76 16.58 ± 1.02 6.80 ± 0.44
#2 7.5 21 21.16 ± 0.63 6.95 ± 0.53 20.96 ± 0.79 6.96 ± 0.57 21.17 ± 0.65 7.23 ± 0.40
#3 8 24 24.47 ± 1.22 8.41 ± 0.72 23.64 ± 1.57 8.22 ± 0.81 23.25 ± 1.43 7.84 ± 0.42
#4 CAM 8.5 27 26.91 ± 0.33 8.24 ± 0.59 26.92 ± 0.34 8.32 ± 0.61 26.93 ± 0.33 8.31 ± 0.39
#5 9 30 30.40 ± 0.38 8.50 ± 0.60 30.41 ± 0.39 8.59 ± 0.63 30.44 ± 0.38 8.83 ± 0.42
#6 9.5 33 32.79 ± 0.46 10.59 ± 0.76 32.81 ± 0.47 10.71 ± 0.79 32.66 ± 0.45 9.62 ± 0.42
#7 MXAM3 10 36 35.77 ± 1.61 9.13 ± 0.76 36.43 ± 1.86 9.39 ± 0.84 37.18 ± 1.81 9.95 ± 0.45
#8 10.5 39 39.26 ± 1.37 9.32 ± 0.73 39.70 ± 1.69 9.53 ± 0.79 40.62 ± 1.87 10.39 ± 0.54
#9 11 42 43.65 ± 2.55 10.63 ± 0.97 44.61 ± 2.52 10.96 ± 1.00 45.43 ± 2.53 11.48 ± 0.85

A summary of burial dose and age estimated from the CAM for De sets from four measured aeolian samples using the MCMC sampling protocol shown in Fig. 2 without and with order constraints. The systematic error for the dose rate measurement shared by all samples was set to σdc=0.1.

Sample Dose rate (Gy/ka) MCMC (without constraints) MCMC (with constraints)
μ (Gy) a (ka) μ (Gy) a (ka)
GL2-1 3.26 ± 0.25 34.55 ± 0.91 10.75 ± 0.96 34.23 ± 0.86 9.72 ± 0.52
GL2-2 2.84 ± 0.21 32.71 ± 1.00 11.68 ± 1.06 32.19 ± 0.92 10.28 ± 0.49
GL2-3 3.23 ± 0.23 30.58 ± 1.04 9.59 ± 0.85 31.03 ± 1.00 10.59 ± 0.52
GL2-4 3.40 ± 0.24 32.24 ± 1.29 9.60 ± 0.85 33.13 ± 1.26 11.13 ± 0.67

Comparisons of burial dose and age estimated from MLE and MCMC for the various statistical age models using De sets from four measured aeolian samples. Quantities estimated using the MCMC sampling protocol shown in Fig. 1 are marked in bold. Quantities estimated using the MLE are inside parentheses.

Sample Dose rate (Gy/ka) CAM MAM3 MXAM3
μ (Gy) a (ka) μ (Gy) a (ka) μ (Gy) a (ka)
GL2-1 3.26 ± 0.25 34.55 ± 0.91 (34.51 ± 0.84) 10.73 ± 0.89 (10.58 ± 0.85) 33.96 ± 1.17 (32.79 ± 3.64) 10.55 ± 0.89 (10.05 ± 1.36) 35.36 ± 1.41 (35.98 ± 3.51) 10.96 ± 0.97 (11.04 ± 1.37)
GL2-2 2.84 ± 0.21 32.70 ± 0.98 (32.65 ± 0.91) 11.64 ± 0.95 (11.50 ± 0.91) 30.40 ± 1.41 (30.33 ± 1.67) 10.83 ± 0.98 (10.68 ± 0.98) 35.53 ± 2.12 (37.12 ± 2.31) 12.66 ± 1.26 (13.7 ± 1.26)
GL2-3 3.23 ± 0.23 30.57 ± 1.05 (30.55 ± 0.98) 9.56 ± 0.77 (9.46 ± 0.74) 27.92 ± 1.55 (26.03 ± 3.56) 8.74 ± 0.80 (8.06 ± 1.24) 35.19 ± 2.03 (36.61 ± 2.41) 11.00 ± 1.01 (11.33 ± 1.10)
GL2-4 3.40 ± 0.24 32.23 ± 1.27 (32.17 ± 1.18) 9.58 ± 0.79 (9.46 ± 0.75) 30.09 ± 1.55 (30.75 ± 1.04) 8.94 ± 0.80 (9.04 ± 0.71) 37.77 ± 2.61 (39.16 ± 2.46) 11.22 ± 1.12 (11.52 ± 1.09)

A summary of Gelman-Rubin convergence diagnostics for measured and simulated samples obtained using the MCMC sampling protocol, as shown in Fig. 2 with order constraints. n.eff is the effective sample size, while Rhat (i.e., the shrink factor) is a statistic measure of the ratio of the average variance of samples within each chain to the variance of the pooled samples across chains. If all chains are at equilibrium, the Rhat will be 1. If these chains have not converged to a common distribution, the Rhat statistic will be greater than 1.

Sample μ a Sample μ a
n_eff Rhat n_eff Rhat n_eff Rhat n_eff Rhat
GL2-1 16000 0.9998 5541.74 1.0002 #1 13865.37 1.0001 9987.93 1.0000
GL2-2 16000 0.9999 8220.18 1.0003 #2 16000 1.0001 11473.55 1.0000
GL2-3 16000 0.9997 7595.60 1.0002 #3 9405.95 1.0004 12466.84 1.0003
GL2-4 16000 0.9998 8864.01 1.0003 #4 16000 1.0003 13161.07 0.9999
#5 16000 0.9999 12830.44 0.9999
#6 16000 0.9999 14048.98 0.9998
#7 16000 0.9999 13073.69 0.9998
#8 7780.82 1.0003 12665.56 1.0001
#9 16000 0.9999 16000 0.9999
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