GLG362/598 Geomorphology

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Simulation of hillslope processes using Microsoft Excel

Purpose: These laboratory exercises are designed to help you gain appreciation for modeling of hillslope processes.

Background information for this exercise comes from (Lecture12.ppt (44 Mb)
Background reading is available from this document: Simple_diffusion_erosion_models.pdf (modified Appendix 1 from Arrowsmith, J R., 1995, Coupled Tectonic Deformation and Geomorphic Degradation along the San Andreas Fault System, Stanford University Ph.D. dissertation 356 pages).

Due: at the beginning of class, October 30, 2006.

Part I: modeling to build intuition


I have prepared this excel file for you: ArrowsmithDiffusionModelingNoData.xls. It is comprised of two sheets: Interface--on which you should do your work; and Model Calculations--second sheet with all of the model calculations so as to make it easier to read. Download it to the folder in which you are working and use it for the exercises.

The variables you have to watch are these: 
These are ones you change:
dx    The horizontal distance between the elevation measurements that make up a profile.
      This can be changed to alter the length of the profile but it is constant for every portion of any profile.
dt    The time between each calculation step.  There are 200 calculation steps in this model.
      You change this number to define the age of the profile (dt*200).
k     This is the transport rate for our assumed transport law.  We discussed independent measurements of this
      parameter that seem to vary generally with current climate such that very dry places like Israel have
      k = 0.1  m2/kyr; the US Basin and Range have k = 1 m2/kyr; and California and Michigan have k = 10 m2/kyr.
Individual change per timestep.  Whatever goes into this cell will be added to the elevation for that
      position each time step.
Initial elevations    Elevations at the beginning of the time of interest that is modeled.


Here ones that are calculated:
H     Final model elevations: Elevations that change as a result of the modeling.
lambda = (k*dt)/(dx2).
      This is a measure of stability for the numerical model and is only for diagnosis purposes.
      It must be <0.5 for the approximations in this modeling to work.
A note on age and t (added 10/24/2006):
If you look at the model calculation space (the second worksheet), you will see that there are 200 time steps, so the dt*200 gives you the age which is what you really want. But we have do to it in terms of dt so that we can keep track of the lambda which is the "speed limit" or the stability measure for the numerical approximations (has to stay below 0.5). Also, if you look at lecture 12 slides 27 and 28, you see that t is used there as the time since zero, or the age, to determine the profile form, but that it is always as a product with k, so that is why the unique form is determined by k*t and so if we know the k*t of the profile and k or t, we can divide through and determine the other (this is what you have to do in part 2 of the exercise).

Exercise tasks

  1. Using the idealized ramp-step initial form (what is initially set up in the Excel Spreadsheet), make a plot showing the initial form and final forms for:
    1. age = 1 ka, and k = 1 m2/kyr
    2. age = 5 ka, and k = 1 m2/kyr
    3. age = 10 ka, and k = 1 m2/kyr
    4. age = 1 ka, and k = 10 m2/kyr.
    Note that age and t are interchangeable here.
    Discuss the effects of varying t and k on the final form. HINT: look at the product k t.
  2. Try different initial profiles. What are some other forms that might be interesting to see what happens as they change shape? Change the bold numbers in the Initial Elevations column and show the development over time of TWO different initial profiles of your choice. Show the profiles at time zero, an intermediate age, and a fairly old and well degraded age. What is the overall character of the change with time? Where does deposition occur and where does erosion occur? Indicate these zones on your plots.
  3. What happens if one of the boundaries for a given initial profile drops at a constant rate? Change the Individual change per timestep column for just the lower or upper end. Be sure to try different downdrop rates. Is the initial form important in the development of the final form?
  4. What happens with a constantly offsetting surface (fault) in the center of the profile? Use the Individual change per timestep and change the lower half of the profile to a small negative number (offset per time step). Be sure to try different offset rates. Note that in reality this could give a problem with slopes > the angle of repose (and violate our assumptions of diffusion erosion).
  5. Try deposition over the lower portion of the profile (similar to above implementation (but with a positive number for the Individual change per timestep for a given cell). What effect on the final form is there from deposition? Be sure to try different deposition rates.

    Part II: applying the model to a real hillslope--morphologic dating of a marine terrace

    Introduction

    Recall the Chris Crosby talk about marine terraces (Marine Terraces Presentation). These terraces formed along the Northern California Coast:

    Overview from Chris. Our profile comes from the whitish box (see below)


    Location of profile that was extracted for this exercise.


    Profiles for exercise. The yellow profile is the observed profile along the line indicated above. The pink line (and the blue one underneath it) shows my estimate of its form at the instant it developed when it was lifted up (or sea level dropped) and removed from wave action.

    I have put the profile into a modified version of the Excel file from above (be careful, it is a fairly large file [3.6 Mb] because of the number of cells required to do the modeling stably): ArrowsmithDiffusionModelingTerraceData.xls. Download this file to the directory in which you are working. Two additions have been made (besides accounting for the larger number of space and time steps). First, I added the observed final profile. Second, we calculate the fit of the model profile to the observed one with this parameter: 
    RMS =  root mean square error
       The equation for this is rms = SQRT(1/p(sum(Hobserved - Hmodel)[summed for all cells in  the entire profile])^2))
       

    
       Where where p is the number of observations, Hi is the observed value at distance xi along the profile, and Hmodel(xi)
       is the model or calculated value at xi.
       It has units of elevation (meters) in this case.  It is a measure of the average distance between the modeled
       elevations and the observed ones.  The best fitting profile is defined as the one which causes the minimum RMS.

    Exercise tasks

    1. Review the table (Hanks, 2002) from Lecture 12 (slide 29) (Lecture12.ppt (44 Mb). Based on what you might know about the region, what do you think will be a good value for k for the northwestern coast of California? Why?
    2. Choose a suitable k, and then progressively change the dt to increase the age. Remember that the morphologic age (kt) is the unique measure of the age. As you increase the morphologic age, write it down along with the corresponding RMS. Make a plot of RMS versus kt:

      It should have the shape of a trough, and the lowest value for RMS will be at the best fitting kt. Choose a few values of kt far from the best fit on both the younger and older sides (but watch out not to increase lambda beyond 0.5 on the spreadsheet), and then focus in on the best fitting value and characterize the curve`s shape there most. What is the best fitting kt? Generally speaking, what is the uncertainty in the fit? Print out the best fitting profile plot and also turn in your plot of RMS versus kt.
    3. At that site, regional correlation indicates that this terrace is from the Stage 5a highstand, or about 83 ka. Assuming that is the age (t), divide your best fitting kt by it to determine a k for this site. What do you get? How does it compare to your estimate from the first question? Why might it be different?

    GLG362/598 Geomorphology
    Last modified: October 22, 2006