Estimating Age-Depth Models from Tracer Contents of Rocks and Sediments

Idea

Assume there is a tracer embedded in rocks or sediment (e.g. Pollen, extraterrestrial 3He, etc.). In addition, there some information on the temporal variations of the tracer available. Then the tracer values measure in the sediment can be compared to the assumed input flux of the tracer to construct rates of sediment accumulation, from which an age-depth model can be constructed.

A standard example is when tracer input into the sediment is constant. The high tracer values reflect low rates of sediment accumulation, which is equivalent to a low gradient of the age-depth model.

This idea is formalized using the idea of stratigraphic content and time content: Stratigraphic content is the tracer value measured in the section (units tracer per length unit), and time content is the input of tracer into the sediment (units tracer per time unit).

Coding tie points

Times and heights of tie points are coded via the functions t_tp (timing) and h_tp (height) that take no inputs. They serve as wrappers around user-defined procedures that reflect uncertainties around tie points. Every time t_tp and h_tp are evaluated, they return possible values for the tie points. Writing these functions requires some effort, but it allows the user to hand over arbitrarily complex uncertainties of the tie points to the strat_cont_to_multiadm function.

The function for the heights of the tie points must return a named vector with entries h1 and h2, corresponding to the heights of the first (lower) and second (upper) tie point. h1 must be smaller than h2.

For this example, I assume the stratigraphic positions of the tie points are known without uncertainty, and are at 10 and 20 m stratigraphic height.

h_min = 10 # stratigraphic height of lower tie point [m]
h_max = 20 # stratigraphic height of upper tie point [m]

h_tp is then implemented as follows:

h_tp = function(){
  return(c("h1" = h_min,
           "h2" = h_max))
}

When evaluated, this function returns the stratigraphic positions of the tie points:

h_tp()
#> h1 h2 
#> 10 20

The function for the timing of the tie points must return a named vector with names “t1” (timing of first tie point) and “t2” (timing of second tie point). “t1” must be smaller than “t2”.

For this example, I assume that the timing of the first tie point follows a normal distribution with mean 0 and standard deviation 0.5. For the second tie point, only maximum and minimum time is available. Due to the lack of information, I assume a uniform distribution between the minimum (9) and the maximum (11). This is implemented as follows in t_tp:

t_tp = function() {
  # timing first tie point
  t1 = rnorm(n = 1, mean = 0, sd = 0.5)
  
  # timing second tie point
  t2 = runif(n = 1, min = 9, max = 11)
  
  return(c("t1" = t1, "t2" = t2))
}

Using Myr as time unit, the distribution of times for the tie points is as follows:

Coding stratigraphic data

Both stratigraphic contents and age contents must be coded as function factories, i.e., functions that return functions. They must be able to take vector inputs and return a vector of the same length as output.

I assume the tracer input into the sediment is constant with time. This is implemented with the following function:

time_cont_gen = function(){
  time_cont = function(x) return(rep(1, length(x)))
  return(time_cont)
}

To visualize this, let’s plot this from time 0 to 10:

# generate one function from time_cont_gen
time_cont = time_cont_gen()
# time where to evaluate the function
t = seq(0,10,by = 0.1)
plot(x = t,
     y = time_cont(t),
     type = "l",
     xlab = "Time [Myr]",
     ylab = "Tracer Input into the Sediment [1/Myr]")

For the tracer measured in the section, I assume it was measured at the bottom, middle, and top of the examined section:

locations = c("h_1" = h_min, "h_2" = mean(c(h_min, h_max)), "h_3" = h_max)

At these locations, tracer values each follow a normal distribution with mean 10, 1, and 8 and standard deviations 1, 0.1, and 0.2.

mean_vals = c("mu_1" = 10, "mu_2" = 1, "mu_3" = 8)
sd_vals = c("sig_1" = 1, "sig_2" = 0.1, "sig_3" = 0.8)

I assume that between the measured points, tracer values change linearly. With these assumptions and values, the function factory for the tracer can be coded as follows

strat_cont_gen = function(){
   loc = locations
   trac_vals = rnorm(n = length(mean_vals),
                     mean = mean_vals,
                     sd = sd_vals)
   strat_cont = approxfun(x = loc,
                          y = trac_vals,
                          yleft = trac_vals[1],
                          yright = trac_vals[3])
   return(strat_cont)
}

To visualize this, here are 3 possible scenarios of how tracer values change in the section:

n = 3 # number of sampled tracer values 
h = seq(h_min, h_max, by = 0.1) # determine age-depth model every 0.1 m
plot(NULL,
     xlim = c(h_min, h_max),
     ylim = c(0, max(mean_vals) +2 * max(sd_vals)),
     xlab = "Stratigraphic Height [m]",
     ylab = "Measured Tracer [1/m]")
cols = c("red", "blue", "black")
for (i in seq_len(n)){
  strat_cont = strat_cont_gen()
  lines(x = h,
        y = strat_cont(h),
        col = cols[i])
}