# Chron.jl

*A Bayesian framework for integrated eruption age and age-depth modelling*

Chron.jl is a two-part framework for any combination of (1) estimating eruption/deposition age distributions from complex mineral age spectra and/or (2) subsequently building a stratigraphic age model based on those distributions. Each step relies on a Markov-Chain Monte Carlo model, and either step can be run as a standalone model if you do not need both components.

The first (distribution) MCMC model is based on the work of Keller, Schoene, and Samperton (2018) and uses information about the possible shape of the true mineral crystallization (or closure) age distribution (e.g., no crystallization possible after eruption or deposition). In this first model, the true eruption or deposition age is a parameter of this scaled crystallization distribution. The stationary distribution of this first MCMC model then gives an estimate of the eruption/deposition age.

The second (stratigraphic) MCMC model, developed for use in Schoene et al. (2019) and Deino et al. (2019) among others, uses the estimated (posterior) eruption/deposition age distributions along with the constraint of stratigraphic superposition to produce an age-depth model. This stratigraphic model can incorporate either standard Gaussian or asymmetric empirical distributions as age constraints, as well additional complications such as hiatuses of known minimum duration, height uncertainty, and one-sided age constraints. The stationary distribution of this second MCMC model yields an estimate of age at each model horizon throughout the section.

In addition to the functions defined and exported here directly, Chron.jl also reexports (and depends upon internally) StatGeochemBase.jl, NaNStatistics.jl, and Isoplot.jl.

For Chron.jl-style age-depth modelling combined with subsidence analysis, see SubsidenceChron.jl

## Citation

You may cite Chron.jl as:

Keller, C.B. (2018).

Chron.jl: A Bayesian framework for integrated eruption age and age-depth modelling. https://doi.org/10.17605/osf.io/TQX3F

Additional citations may include: For eruption age estimation

Keller, C.B., Schoene, B., & Samperton, K.M. (2018). A Stochastic Sampling Approach to Zircon Eruption Age Interpretation.

Geochemical Persectives Letters8, 31–35

For the extension of this eruption age estimation to sanidine Ar-Ar data

van Zalinge, M.E, Mark, D.F., Sparks R.S.J., Tremblay, M.M. Keller, C.B., Cooper, F.J. & Rust, A. (2022). Timescales for pluton growth, magma-chamber formation and super-eruptions.

Nature608, 87-92.

For age-depth modelling, applied to zircon U-Pb data

Schoene, B., Eddy, M.P., Samperton, K.M., Keller, C.B., Keller, G., Adatte, T., & Khadri, S.F.R. (2019). U-Pb constraints on pulsed eruption of the Deccan Traps across the end-Cretaceous mass extinction.

Science363 (6429), 862–866.

For age-depth modelling, applied to sanidine Ar-Ar data

Deino, A.L., Dommain, R., Keller, C.B., Potts, R., Behrensmeyer, A.K., Beverly, E.J., King, J., Heil, C.W., Stockhecke, M., Brown, E.T., Moerman, J., de Menocal, P., Levin, N.E., & ODP Scientific Team. (2019). Chronostratigraphic model of a high-resolution drill core record of the past million years from the Koora Basin, south Kenya Rift: Overcoming the difficulties of variable sedimentation rate and hiatuses.

Quaternary Science Reviews215, 213–231.

## Installation

Chron.jl is written in the Julia programming language, and is registered on the General registry. To install, enter the Julia package manager (type ] in the REPL) and type:

```
pkg> add Chron
```

If you are trying to use Chron with a published script written prior to ~2021, you may want to use the oldest registered version of the package, which you can install with (e.g.)

```
pkg> add Chron@v0.1
```

## Online / Notebook Usage

### Coupled eruption/deposition age and age-depth modelling

For a quick test (without having to install anything), try the interactive online Jupyter notebook (note: it'll take a few minutes for the notebook to launch).

This runs examples/Chron1.0Coupled.ipynb on a JupyterHub server hosted by the Binder project. If you make changes to the interactive online notebook, you can save them with `File`

> `Download as`

> `Notebook (.ipynb)`

To run a downloaded notebook locally, use IJulia

```
julia> using IJulia
julia> notebook()
```

For an example of the Pb-loss-aware options, see also

### Standalone age-depth modelling

If you want to use Chron.jl for for age-depth modelling without the eruption/deposition age estimation step, there are also example notebooks standalone age-depth modelling using either

- Age-depth modelling with simple Gaussian age constraints or
- Age-depth modelling with radiocarbon age constraints

with or without hiatuses.

### Standalone eruption/deposition age modelling

To run an eruption/deposition age estimate, without any age-depth modelling, try the notebook for

## Standard Usage

After installing Julia with or without Juno, and Chron.jl (above), run examples/Chron1.0Coupled.jl to see how the code works. It should look something like this:

#### Load necessary Julia packages

if VERSION>=v"0.7"
using Statistics, StatsBase, DelimitedFiles, SpecialFunctions
else
using Compat
end
using Chron
using Plots; gr();

#### Enter sample information

This example data is from Clyde et al. (2016) "Direct high-precision U–Pb geochronology of the end-Cretaceous extinction and calibration of Paleocene astronomical timescales" EPSL 452, 272–280. doi: 10.1016/j.epsl.2016.07.041

nSamples = 5 # The number of samples you have data for
smpl = ChronAgeData(nSamples)
smpl.Name = ("KJ08-157", "KJ04-75", "KJ09-66", "KJ04-72", "KJ04-70",)
smpl.Height[:] = [ -52.0, 44.0, 54.0, 82.0, 93.0,]
smpl.Height_sigma[:] = [ 3.0, 1.0, 3.0, 3.0, 3.0,]
smpl.Age_Sidedness[:] = zeros(nSamples) # Sidedness (zeros by default: geochron constraints are two-sided). Use -1 for a maximum age and +1 for a minimum age, 0 for two-sided
smpl.Path = "DenverUPbExampleData/" # Where are the data files?
smpl.inputSigmaLevel = 2 # i.e., are the data files 1-sigma or 2-sigma. Integer.
AgeUnit = "Ma" # Unit of measurement for ages and errors in the data files
HeightUnit = "cm"; # Unit of measurement for Height and Height_sigma

For each sample in `smpl.Name`

, we must have a `.csv`

file in `smpl.Path`

which contains each individual mineral age and uncertainty. For instance, examples/DenverUPbExampleData/KJ08-157.csv contains:

```
66.12,0.14
66.115,0.048
66.11,0.1
66.11,0.17
66.096,0.056
66.088,0.081
66.085,0.076
66.073,0.084
66.07,0.11
66.055,0.043
66.05,0.16
65.97,0.12
```

Note also that smpl.Height *must* increase with increasing stratigraphic height -- i.e., stratigraphically younger samples must be more positive. For this reason, it is convenient to represent depths below surface as negative numbers.

#### Configure and run eruption/deposition age model

To learn more about the eruption/deposition age estimation model, see also Keller, Schoene, and Samperton (2018) and the BayeZirChron demo notebook. It is important to note that this model (like most if not all others) has no knowledge of open-system behaviour, so *e.g.*, Pb-loss will lead to erroneous results.

# Number of steps to run in distribution MCMC
distSteps = 10^7
distBurnin = floor(Int,distSteps/100)
# Choose the form of the prior distribution to use.
# A variety of potentially useful distributions are provided in DistMetropolis.jl - Options include UniformDisribution,
# TriangularDistribution, BootstrappedDistribution, and MeltsVolcanicZirconDistribution - or you can define your own.
dist = TriangularDistribution;
# Run MCMC to estimate saturation and eruption/deposition age distributions
smpl = tMinDistMetropolis(smpl,distSteps,distBurnin,dist);

```
Estimating eruption/deposition age distributions...
1: KJ08-157
2: KJ04-75
3: KJ09-66
4: KJ04-72
5: KJ04-70
```

Let's see what that did

; ls $(smpl.Path)
results = readdlm(smpl.Path*"results.csv",',')
; open $(smpl.Path*"KJ04-75_rankorder.pdf")

```
BootstrappedDistribution.pdf
KJ04-70.csv
KJ04-70_distribution.pdf
KJ04-70_rankorder.pdf
KJ04-72.csv
KJ04-72_distribution.pdf
KJ04-72_rankorder.pdf
KJ04-75.csv
KJ04-75_distribution.pdf
KJ04-75_rankorder.pdf
KJ08-157.csv
KJ08-157_distribution.pdf
KJ08-157_rankorder.pdf
KJ09-66.csv
KJ09-66_distribution.pdf
KJ09-66_rankorder.pdf
KJ12-01.csv
results.csv
6×5 Array{Any,2}:
"Sample" "Age" "2.5% CI" "97.5% CI" "sigma"
"KJ08-157" 66.065 66.0312 66.0896 0.0151996
"KJ04-75" 65.9744 65.9237 66.0056 0.0198365
"KJ09-66" 65.9475 65.9143 65.9807 0.0168379
"KJ04-72" 65.9531 65.9194 65.9737 0.0135548
"KJ04-70" 65.8518 65.7857 65.898 0.0288371
```

Let's look at the plots for sample KJ04-70:

For each sample, the eruption/deposition age distribution is inherently asymmetric, because of the one-sided relationship between mineral closure and eruption/deposition. For example:

Consequently, rather than simply taking a mean and standard deviation of the stationary distribtuion of the Markov Chain, the histogram of the stationary distribution is fit to an empirical distribution function of the form

where

*i.e.*, an asymmetric exponential function with two log-linear segments joined with an arctangent sigmoid. After fitting, the five parameters $a$ - $e$ are stored in `smpl.params`

and passed to the stratigraphic model

#### Configure and run stratigraphic model

For a publication-quality result, you probably want nsteps and burnin on the order of

```
config.nsteps = 30000 # Number of steps to run in distribution MCMC
config.burnin = 10000*npoints_approx # Number to discard
```

and examine the log likelihood plot to make sure you've converged.

To run the stratigraphic MCMC model, we call the `StratMetropolisDist`

function. If you want to skip the first step and simply input run the stratigraphic model with Gaussian mean age and standard deviation for some number of stratigraphic horizons, then you can set `smpl.Age`

and `smpl.Age_sigma`

directly, but then you'll need to call `StratMetropolis`

instead of `StratMetropolisDist`

# Configure the stratigraphic Monte Carlo model
config = StratAgeModelConfiguration()
# If you in doubt, you can probably leave these parameters as-is
config.resolution = 1.0 # Same units as sample height. Smaller is slower!
config.bounding = 0.1 # how far away do we place runaway bounds, as a fraction of total section height
(bottom, top) = extrema(smpl.Height)
npoints_approx = round(Int,length(bottom:config.resolution:top) * (1 + 2*config.bounding))
config.nsteps = 15000 # Number of steps to run in distribution MCMC
config.burnin = 10000*npoints_approx # Number to discard
config.sieve = round(Int,npoints_approx) # Record one out of every nsieve steps
# Run the stratigraphic MCMC model
(mdl, agedist, lldist) = StratMetropolisDist(smpl, config); sleep(0.5)
# Plot the log likelihood to make sure we're converged (n.b burnin isn't recorded)
plot(lldist,xlabel="Step number",ylabel="Log likelihood",label="",line=(0.85,:darkblue))=

```
Generating stratigraphic age-depth model...
Burn-in: 1750000 steps
Collecting sieved stationary distribution: 2625000 steps
```

#### Plot results

# Plot results (mean and 95% confidence interval for both model and data)
hdl = plot([mdl.Age_025CI; reverse(mdl.Age_975CI)],[mdl.Height; reverse(mdl.Height)], fill=(minimum(mdl.Height),0.5,:blue), label="model")
plot!(hdl, mdl.Age, mdl.Height, linecolor=:blue, label="", fg_color_legend=:white)
plot!(hdl, smpl.Age, smpl.Height, xerror=(smpl.Age-smpl.Age_025CI,smpl.Age_975CI-smpl.Age),label="data",seriestype=:scatter,color=:black)
plot!(hdl, xlabel="Age ($AgeUnit)", ylabel="Height ($HeightUnit)")
savefig(hdl,"AgeDepthModel.pdf");
display(hdl)

# Interpolate results at KTB (height = 0)
height = 0
KTB = linterp1s(mdl.Height,mdl.Age,height)
KTB_min = linterp1s(mdl.Height,mdl.Age_025CI,height)
KTB_max = linterp1s(mdl.Height,mdl.Age_975CI,height)
print("Interpolated age: $KTB +$(KTB_max-KTB)/-$(KTB-KTB_min) Ma")
# We can also interpolate the full distribution:
interpolated_distribution = Array{Float64}(undef,size(agedist,2))
for i=1:size(agedist,2)
interpolated_distribution[i] = linterp1s(mdl.Height,agedist[:,i],height)
end
histogram(interpolated_distribution, xlabel="Age (Ma)", ylabel="N", label="", fill=(0.85,:darkblue), linecolor=:darkblue)

```
Interpolated age: 66.01580546918152 +0.04924877964148777/-0.049571492234548487 Ma
```

There are other things we can plot as well, such as deposition rate:

# Set bin width and spacing
binwidth = 0.01 # Myr
binoverlap = 10
ages = collect(minimum(mdl.Age):binwidth/binoverlap:maximum(mdl.Age))
bincenters = ages[1+Int(binoverlap/2):end-Int(binoverlap/2)]
spacing = binoverlap
# Calculate rates for the stratigraphy of each markov chain step
dhdt_dist = Array{Float64}(undef, length(ages)-binoverlap, config.nsteps)
@time for i=1:config.nsteps
heights = linterp1(reverse(agedist[:,i]), reverse(mdl.Height), ages)
dhdt_dist[:,i] = abs.(heights[1:end-spacing] - heights[spacing+1:end]) ./ binwidth
end
# Find mean and 1-sigma (68%) CI
dhdt = nanmean(dhdt_dist,dim=2)
dhdt_50p = nanmedian(dhdt_dist,dim=2)
dhdt_16p = nanpctile(dhdt_dist,15.865,dim=2) # Minus 1-sigma (15.865th percentile)
dhdt_84p = nanpctile(dhdt_dist,84.135,dim=2) # Plus 1-sigma (84.135th percentile)
# Other confidence intervals (10:10:50)
dhdt_20p = nanpctile(dhdt_dist,20,dim=2)
dhdt_80p = nanpctile(dhdt_dist,80,dim=2)
dhdt_25p = nanpctile(dhdt_dist,25,dim=2)
dhdt_75p = nanpctile(dhdt_dist,75,dim=2)
dhdt_30p = nanpctile(dhdt_dist,30,dim=2)
dhdt_70p = nanpctile(dhdt_dist,70,dim=2)
dhdt_35p = nanpctile(dhdt_dist,35,dim=2)
dhdt_65p = nanpctile(dhdt_dist,65,dim=2)
dhdt_40p = nanpctile(dhdt_dist,40,dim=2)
dhdt_60p = nanpctile(dhdt_dist,60,dim=2)
dhdt_45p = nanpctile(dhdt_dist,45,dim=2)
dhdt_55p = nanpctile(dhdt_dist,55,dim=2)
# Plot results
hdl = plot(bincenters,dhdt, label="Mean", color=:black, linewidth=2)
plot!(hdl,[bincenters; reverse(bincenters)],[dhdt_16p; reverse(dhdt_84p)], fill=(minimum(mdl.Height),0.2,:darkblue), linealpha=0, label="68% CI")
plot!(hdl,[bincenters; reverse(bincenters)],[dhdt_20p; reverse(dhdt_80p)], fill=(minimum(mdl.Height),0.2,:darkblue), linealpha=0, label="")
plot!(hdl,[bincenters; reverse(bincenters)],[dhdt_25p; reverse(dhdt_75p)], fill=(minimum(mdl.Height),0.2,:darkblue), linealpha=0, label="")
plot!(hdl,[bincenters; reverse(bincenters)],[dhdt_30p; reverse(dhdt_70p)], fill=(minimum(mdl.Height),0.2,:darkblue), linealpha=0, label="")
plot!(hdl,[bincenters; reverse(bincenters)],[dhdt_35p; reverse(dhdt_65p)], fill=(minimum(mdl.Height),0.2,:darkblue), linealpha=0, label="")
plot!(hdl,[bincenters; reverse(bincenters)],[dhdt_40p; reverse(dhdt_60p)], fill=(minimum(mdl.Height),0.2,:darkblue), linealpha=0, label="")
plot!(hdl,[bincenters; reverse(bincenters)],[dhdt_45p; reverse(dhdt_55p)], fill=(minimum(mdl.Height),0.2,:darkblue), linealpha=0, label="")
plot!(hdl,bincenters,dhdt_50p, label="Median", color=:grey, linewidth=1)
plot!(hdl, xlabel="Age ($AgeUnit)", ylabel="Depositional Rate ($HeightUnit / $AgeUnit over $binwidth $AgeUnit)", fg_color_legend=:white)
# savefig(hdl,"DepositionRateModelCI.pdf")
display(hdl)

### Stratigraphic model including hiatuses

We can also deal with discrete hiatuses in the stratigraphic section if we know where they are and about how long they lasted. We need some different models and methods though. In particular, in addition to the `StratAgeData`

struct, we also need a `HiatusData`

struct now, and we're going to want to pass these to `StratMetropolisDistHiatus`

instead of `StratMetropolisDist`

like before.

# Data about hiatuses
nHiatuses = 2 # The number of hiatuses you have data for
hiatus = HiatusData(nHiatuses) # Struct to hold data
hiatus.Height = [20.0, 35.0 ]
hiatus.Height_sigma = [ 0.0, 0.0 ]
hiatus.Duration = [ 0.2, 0.43]
hiatus.Duration_sigma = [ 0.05, 0.07]
# Run the model. Note: we're using `StratMetropolisDistHiatus` now, instead of just `StratMetropolisDistHiatus`
(mdl, agedist, hiatusdist, lldist) = StratMetropolisDistHiatus(smpl, hiatus, config); sleep(0.5)
# Plot results (mean and 95% confidence interval for both model and data)
hdl = plot([mdl.Age_025CI; reverse(mdl.Age_975CI)],[mdl.Height; reverse(mdl.Height)], fill=(minimum(mdl.Height),0.5,:blue), label="model")
plot!(hdl, mdl.Age, mdl.Height, linecolor=:blue, label="", fg_color_legend=:white)
plot!(hdl, smpl.Age, smpl.Height, xerror=(smpl.Age-smpl.Age_025CI,smpl.Age_975CI-smpl.Age),label="data",seriestype=:scatter,color=:black)
plot!(hdl, xlabel="Age (Ma)", ylabel="Height (cm)")

```
Generating stratigraphic age-depth model...
Burn-in: 1750000 steps
Collecting sieved stationary distribution: 2625000 steps
```