A model framework for interpretation of mineral age spectra in stratigraphic context
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README.md

Chron.jl

DOI

A two-part framework for (1) estimating eruption/deposition age distributions from complex mineral age spectra and (2) subsequently building a stratigraphic age model based on those distributions. Each step relies on a Markov-Chain Monte Carlo model.

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 uses the estimated (posterior) eruption/deposition age distributions along with the constraint of stratigraphic superposition to produce an age-depth model

Installation

Chron.jl is written in the Julia programming language.

In the Julia package manager (type ] in the REPL)

(v1.0) pkg> add "https://github.com/brenhinkeller/Chron.jl"

or for previous versions of Julia, in the REPL

julia> Pkg.clone("https://github.com/brenhinkeller/Chron.jl")

Usage

Online / notebook usage

Binder

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/demo.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()

Standard usage

After installing Julia with or without Juno, and Chron.jl (above), run examples/demo.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 = NewStratAgeData(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

Configure and run eruption/deposition age model

To learn more about the eruption/deposition age estimation model, see also Keller, Schoene, and Sameperton (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:

svg

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:

svg

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

note: to spare Binder's servers, this demo uses

config.nsteps = 3000
config.burnin = 2000*npoints_approx

However, you probably want higher numbers for a publication-quality result, for instance

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 = NewStratAgeModelConfiguration()
# 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

png

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)

png

# 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)

png

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 = pctile(dhdt_dist,15.865,dim=2) # Minus 1-sigma (15.865th percentile)
dhdt_84p = pctile(dhdt_dist,84.135,dim=2) # Plus 1-sigma (84.135th percentile)
# Other confidence intervals (10:10:50)
dhdt_20p = pctile(dhdt_dist,20,dim=2)
dhdt_80p = pctile(dhdt_dist,80,dim=2)
dhdt_25p = pctile(dhdt_dist,25,dim=2)
dhdt_75p = pctile(dhdt_dist,75,dim=2)
dhdt_30p = pctile(dhdt_dist,30,dim=2)
dhdt_70p = pctile(dhdt_dist,70,dim=2)
dhdt_35p = pctile(dhdt_dist,35,dim=2)
dhdt_65p = pctile(dhdt_dist,65,dim=2)
dhdt_40p = pctile(dhdt_dist,40,dim=2)
dhdt_60p = pctile(dhdt_dist,60,dim=2)
dhdt_45p = pctile(dhdt_dist,45,dim=2)
dhdt_55p = pctile(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)

png

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 = NewHiatusData(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

png