Bias of lag times derived from the inversion of bimodal closure age distributions

In this example, we will use a true parent closure age distribution which is bimodal with a first and second true closure peak age. We will be inverting for closure age models using the binomfit routines. For this input closure age model only 1 or 2 peaks are neccessary to explain the data and in practice the BIC does not prefer inversions with more peaks.

We use either 51 for 102 grains where the nI values are taken from one of Hugh's Alpine samples. True closure ages are randomly assigned to each grain.

The important result that this analysis shows is that if we specify a constant closure age lag model - the P1 component ages will be biased to older ages for absolutely older models - which would naively be interpreted as a reduction in the lag time for younger samples where, we don't actually know the true closure age independently. This is because single age models are prefered for absolute older ages and the P1 age in these models is a central age.

Proportion of inverted results with 1 and 2 closure ages prefered

To start the analysis, I have swept through a large range of models with different first component ages and different lags. I have run 100 random models for each age pair so that we can look at the range of inverted models consistent with a single true closure age model where only stochastic uncertainty in the number of spontaneous tracks is affecting the result.

Below, I have analysed how the BIC prefered of age components in the inverted model varies with the true closure ages with the younger closure plotted on the bottom axis and the older closure age plotted on the y-axis either as an absolute age in the lower figures or as a relative true lag age in the upper figures; the lag plot is clearer.

Each point in this plot represents an average over 100 simulations. The colour shows the proportion of the inversion which prefered for the correct 2 age model (in green) or a single component age (red). Unsurprisingly, shorter lags can be resolved at younger ages.

The right and left columns show the results for 51 and 102 grains. The best fit line indicates a relatively linear portion of the transition between 1 and 2 age models. This might be analytically tractable - but I have not managed it…!

Importantly, if we are considering a constant lag time of, for example, 10Myr we will transition from prefering a 2 component age inversion to a 1 component age inversion at ~15-20Mry. We explore the significance of this in the next section

multiplot(P1, P2, P3, P4, cols = 2)

## Loading required package: grid

summary(fit1)

## 
## Call:
## lm(formula = yHalf[40:120] ~ seq(40, 120))
## 
## Residuals:
##    Min     1Q Median     3Q    Max 
## -1.845 -0.371  0.045  0.395  3.577 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   9.97441    0.33518    29.8   <2e-16 ***
## seq(40, 120)  0.25325    0.00402    63.0   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.846 on 79 degrees of freedom
## Multiple R-squared:  0.98,   Adjusted R-squared:  0.98 
## F-statistic: 3.97e+03 on 1 and 79 DF,  p-value: <2e-16

summary(fit2)

## 
## Call:
## lm(formula = yHalf[40:120] ~ seq(40, 120))
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -1.6305 -0.3221  0.0207  0.3807  1.2437 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept)   7.51671    0.21425    35.1   <2e-16 ***
## seq(40, 120)  0.23428    0.00257    91.1   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.541 on 79 degrees of freedom
## Multiple R-squared:  0.991,  Adjusted R-squared:  0.99 
## F-statistic: 8.31e+03 on 1 and 79 DF,  p-value: <2e-16

Analysis of a constant 10Ma lag closure age model

Here we investigate the prefered P1 ages for the case with 102 grains above using a constant true lag of 10 Ma. This corresponds to a horizontal slice through figure © above. Notice that we switch from prefering the correct 2 component age model to a single component age model at ~15Ma; though this transition is not sharp.

In particular, we are intersted in how the differece between the P1 component and the true first component age vary with the absolute age. Consequently, I have plotted histograms of the 100 prefered P1 component ages below with the true age of the first peak indicated in the sub figure title.

The red and green lines show the first and second component ages respectively which are separated by 10Ma. The blue dashed line shows the median P1 age. Primarily, we are interested in the difference between the red and blue lines.

Absolutely Young Ages

• At 2Ma, the P1 ages are roughly symmetical about the true first age.
• As the first age gets older, we see a progressive switch to an asymmetric distribution and the median P1 age shifts to older ages relative to the first component.
• The bimodal age distribution most clearly seen with fist ages of 15 and 21 Ma derives from a preference for both the single and double peak models in the random sampling.

Absolutely Old Ages

• At old first peak ages, the single model is srongly prefered
  • Consequently the P1 age is more representative of a central age which, given a lag of 10Ma, the central age will be on average 5Ma older than the first peak.

Conslusions

So my interpretation of this synthetic is:

• If we did not know the true closure age model in advance, we would observe a constant lag at older times and a reduction in the lag at short times.
• The older lag would be artificaially old as the technique is not capable of resolving lags of 10Ma at ages greater than ~20Ma for the grain distributions we have.
• There is the potential to interpret an acceleration in erosion due to this artefact.
• Synthetic analyses such as performed here can be helpful in constraining whether a binomfit analysis is likely to contain such artefacts.

## Error: invalid type (NULL) for variable 'yHalf[40:120]'