# Reconstructing $\delta\mathrm{D}_{p}$ from $\delta\mathrm{D}_{wax}$

# Introduction

Water isotopes of precipitation record changes in temperature, precipitation, atmospheric circulation, and many other physical processes. This utility has led to wide adoption in the geosciences to understand Earth’s history. I’m a paleoclimatologist interested how past atmospheric circulation changes produce climate changes and so I’ve used the isotopic composition of leaf waxes to understand atmospheric circulation in the tropical Pacific. Often, it’s useful to move from $\delta\mathrm{D}_{wax}$ to $\delta\mathrm{D}_{p}$ for comparison against isotope-enabled climate simulations and the contemporary climate. This enables powerful proxy-model comparisons that can shed light on the mechanisms of past climate changes.

Moving from $\delta\mathrm{D}_{wax}$ to $\delta\mathrm{D}_{p}$ requires accounting for the biological fractionation $\epsilon_{p}$ that takes place when a plant incorporates meteoric waters into its tissues. Recently, a paper developed a Bayesian approach to reconstruct $\delta\mathrm{D}_{p}$ from $\delta\mathrm{D}_{wax}$ by leveraging the strength of contemporary measurements of the carbon isotopic composition of plant tissues. Making use of this statistical approach requires having paired measurements of $\delta\mathrm{D}_{wax}$ and $\delta^{13}\mathrm{C}_{wax}$ from a sedimentary archive.

In this example, I’m using data from the paper that first formalized this statistical approach - Tierney et al., (2017, Science Advances). This approach has since been used a variety of studies that use $\delta\mathrm{D}_{wax}$. Importantly, this procedure assumes that vegetation is largely unchanged over the duration of the time series which can be assessed by plotting the $\delta^{13}\mathrm{C}_{wax}$ against time. The key feature we are looking for is nearly constant $\delta^{13}\mathrm{C}_{wax}$ values through time which ensures that we have a unchanging plant photosynthetic pathway and $\epsilon_{p}$. If this requirement is meant, this Bayesian approach is appropriate to use to reconstruction $\delta\mathrm{D}_{p}$.

# Bayesian Approach

See Tierney et al., (2017, Science Advances) for the mathematical basis, but briefly the equations used here:

**Equations to calculate $\delta\mathrm{D}_{p}$**

(1) $\delta D_{p} = \frac{1000 + \delta D_{wax}}{\frac{\epsilon_{p}}{1000} + 1} - 1000$

(2) $\epsilon_{p} = f_C4 \times \epsilon_{C4} + (1 - f_{C4}) \times \epsilon_{C3}$

(3) $Y = (\frac{\delta^{13}C_{wax} - \delta^{13}C_{C3}}{\delta^{13}C_{C4} - \delta^{13}C_{C3}}) \times N$

**Prior distribution for model parameters ($\theta$; i.e. $f_{C4}$):**

$p(\theta) = \theta^{\alpha - 1}(1 - \theta)^{\beta - 1}$

**Posterior distribution which we will be sampling:**

$p(\theta | Y) \propto \theta^{Y + \alpha - 1} (1 - \theta)^{N - Y + \beta - 1}$

We will be assuming an uninformed prior, hence $\alpha = 1$ and $\beta = 1$

# Resampling Algorithm

- Sample $C_3$ and $C_4$ $\delta^{13}\mathrm{C}$ end-member values. Here, taken from Garcin et al., (2014, Geochim. Cosmochim. Acta).

```
# First set N - the number of plants contributing to the observed measurement,
# assumed to be large here
N <- 5000
# Value from Garcin et al., (2014, Geochim. Cosmochim. Acta)
d13c_c4_mean <- -19.8
d13c_c4_sd <- 0.4
# Value from Garcin et al., (2014, Geochim. Cosmochim. Acta)
d13c_c3_mean <- -33.4
d13c_c3_sd <- 0.4
# Sample from a normal distribution
d13c_c4_sample <- rnorm(N, mean = d13c_c4_mean, sd = d13c_c4_sd)
d13c_c3_sample <- rnorm(N, mean = d13c_c3_mean, sd = d13c_c3_sd)
```

- Calculate $Y$ using Equation 3.

```
y = ((D13C_WAX_VALUE_GOES_HERE - d13c_c3_sample)/(d13c_c4_sample - d13c_c3_sample)) * N
```

- Sample the posterior beta distribution.

```
# Sample posterior for fraction of C4 plants
# Using sapply here because we want a beta distribution sample for each of the N
# values we calculated in Step 2.
f_c4 <- sapply(Y, function(y) rbeta(1, shape1 = y + 1 - 1, shape2 = N - y + 1 - 1))
```

- Monte-Carlo resampling to propagate errors from $\epsilon_{C3}$, $\epsilon_{C4}$, and $\delta D_{wax}$ measurements and calculate $\epsilon_{p}$ from Equation 2.

```
# Value from Sachse et al., (2012, Annu. Rev. Earth Planet. Sci.)
epsilon_c4_mean <- -126
epsilon_c4_sd <- 4
# Value from Sachse et al., (2012, Annu. Rev. Earth Planet. Sci.)
epsilon_c3_mean <- -113
epsilon_c3_sd <- 2
# Sample epsilon value end members from Sachse et al., (2012, Annu. Rev. Earth Planet. Sci.)
epsilon_c4_sample <- rnorm(1, mean = epsilon_c4_mean, sd = epsilon_c4_sd)
epsilon_c3_sample <- rnorm(1, mean = epsilon_c3_mean, sd = epsilon_c3_sd)
# Sample dD of sample - assuming 2 per mil standard deviation for analytical
# uncertainty.
dD_sample <- rnorm(N, mean = DELTA_D_WAX_VALUE_GOES_HERE, sd = 2)
# Calculate epsilon_p
epsilon_p <- f_c4 * epsilon_c4_sample + (1 - f_c4) * epsilon_c3_sample
```

- With all model parameters in place, calculate $\delta D_{p}$ from $\delta D_{wax}$ using Equation 1.

```
ensemble <- ((1000 + dD_sample) / ((epsilon_p/1000) + 1)) - 1000
```

As written, this code will provide an ensemble of 5,000 estimates of $\delta D_{p}$ for a single pair of $\delta D_{wax}$ and $\delta^{13}C_{wax}$ measurements. In practice, that’s now how we want this code to work. We want to feed in an entire spreadsheet and have the results exported in another spreadsheet. Below is a code chunk that can make this happen with data from Tierney et al., (2017, Science Advances).

# Application

Let’s apply this algorithm and statistics to reproduce the results from Tierney et al., (2017, Science Advances)

```
# Setting seed to make sure the random numbers stay constant for this script
set.seed(1)
# First set N - the number of plants contributing to the observed measurement,
# assumed to be large here.
N <- 5000
# Define the number of iterations (i.e. resamples). Note, I am doing all
# calculations in vector form here, so I never iterate i to the end of iter.
# Rather, I use `iter` to define the number of draws from each distribution
iter <- 1000
# Value from Garcin et al., (2014, Geochim. Cosmochim. Acta)
d13c_c4_mean <- -19.8
d13c_c4_sd <- 0.4
# Value from Garcin et al., (2014, Geochim. Cosmochim. Acta)
d13c_c3_mean <- -33.4
d13c_c3_sd <- 0.4
# Value from Sachse et al., (2012, Annu. Rev. Earth Planet. Sci.)
epsilon_c4_mean <- -126
epsilon_c4_sd <- 4
# Value from Sachse et al., (2012, Annu. Rev. Earth Planet. Sci.)
epsilon_c3_mean <- -113
epsilon_c3_sd <- 2
# Read in data
gc27 <- read.table("~/Downloads/tierney2017gc27.txt", skip = 111, header = TRUE)
# Prepare output list - it will be structured as a list of lists with entries
# 1 to the length of the gc27$dDwax_iv corresponding to each dD_wax sample.
output <- list(
delta_D_precip = list(),
f_c4 = list(),
epsilon_precip = list()
)
# Iterate through each ice-corrected measurement of delta D_{wax}
for(i in 1:length(gc27$dDwax_iv)){
# Sample d13C end members from Garcin et al., (2014, Geochim. Cosmochim. Acta)
d13c_c4_sample <- rnorm(iter, mean = d13c_c4_mean, sd = d13c_c4_sd)
d13c_c3_sample <- rnorm(iter, mean = d13c_c3_mean, sd = d13c_c3_sd)
# Sample epsilon value end members from Sachse et al., (2012, Annu. Rev. Earth Planet. Sci.)
epsilon_c4_sample <- rnorm(iter, mean = epsilon_c4_mean, sd = epsilon_c4_sd)
epsilon_c3_sample <- rnorm(iter, mean = epsilon_c3_mean, sd = epsilon_c3_sd)
# Sample dD of sample
dD_sample <- rnorm(iter, mean = gc27$dDwax_iv[i], sd = 2)
# Calculate y
Y = ((rep(gc27$d13Cwax[i], iter) - d13c_c3_sample)/(d13c_c4_sample - d13c_c3_sample)) * N
# Sample posterior for fraction of C4 plants
f_c4 <- sapply(Y, function(y) rbeta(1, shape1 = y + 1 - 1, shape2 = N - y + 1 - 1))
# Monte carlo resampling for epsilon_p
epsilon_p <- f_c4 * epsilon_c4_sample + (1 - f_c4) * epsilon_c3_sample
# Monte Carlon resampling for dD_precip
delta_D_precip_ensemble <- ((1000 + dD_sample) / ((epsilon_p/1000) + 1)) - 1000
# Writing to our output list
output$delta_D_precip[[i]] <- delta_D_precip_ensemble
output$f_c4[[i]] <- f_c4
output$epsilon_precip[[i]] <- epsilon_p
}
# Let's get summary statistics from our output list
results <- data.frame(
dDP = sapply(X = output$delta_D_precip, FUN = mean),
dDP_1s_lower = sapply(X = output$delta_D_precip, FUN = function(x) mean(x) - sd(x)),
dDP_1s_lower = sapply(X = output$delta_D_precip, FUN = function(x) mean(x) + sd(x))
)
# And compare against the results that are within the Tierney et al., (2017,
# Science Advances) data file.
mean(gc27$dDP - results$dDP)
# The difference is 0.01968102 - so its safe to say we've replicated the results
```

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