Modeling a graph that shows age in relation to depth of an ice sample

[hey123]

Homework Statement

The thickness of the annually deposited ice layers in the ice core can be determined by examining chemical impurities and isotope ratios. Under pressure, ice exhibits the properties of a liquid. Therefore, the ice in the Greenland ice sheet is not only compressed, but also moves from the ice shelf towards the coasts. As a result, the layers of ice deposited each year become thinner with increasing depth. The following table shows the thickness of the ice layers in meters per year for the NEEM drill core:

Depth z in m 0 500 1000 1200 1400 1500 1600 Thickness per year λ in m a-1 0.25 0.20 0.13 0.10 0.037 0.018 0.010

Using the data from the table, create a graph that approximately shows the age t in a of an ice sample as a function of the depth z at which the sample was taken.

Relevant Equations

a= m* a/m, i.e. t=z/λ.

To solve this, I first used the units to work out that a= m* a/m, i.e. t=z/λ. This would allow you to determine the time duration within an interval section by section and then add this to the previous ones to obtain the age of the respective layer. However, this would require a constant thickness per year for each interval. However, since this is most likely not the case, my next consideration was that the age must be the integral of a 1/λ(z) function, which I cannot model.

 

[pasmith]

You can use linear interpolation between the data points. Between z_i and z_{i+1} that gives you \int_{z_i}^z \frac{1}{\lambda(z)}\,dz = \int_{z_i}^z \frac{1}{A_i + B_iz}\,dz which you can do analytically.