1. Geology Geomath
Estimating the coefficients of linear, exponential, polynomial, logarithmic, and power law expressions
tom.h.wilson
Department of Geology and Geography
West Virginia University
Morgantown, WV
Tom Wilson, Department of Geology and Geography

2. Objectives -
Show how the computer can be used to estimate the coefficients of various quantitative relationships in geology.
These include:
the linear age-depth relationship discussed by Waltham
the exponential porosity depth relationship
polynomial relationship between temperature and depth
and general power law relationships such as the Gutenberg-Richter relation
We’ll look at some familiar relationships that we’ve already worked with
Tom Wilson, Department of Geology and Geography

3. Problem 2.15
The rate of accumulation, p, of carbonate sediments on a reef is given approximately by
Po is the initial accumulation rate and Z is the depth at which the accumulation rate drops to 0.37 of its initial value
What actually goes on tends to be more complicate din this case. So sometimes we simplify or generalize our representations
Tom Wilson, Department of Geology and Geography

4. Problem 4.7 (Any questions for today?)
The thickness of a bottomset bed at the foot of a delta can often be well approximated by
Where t is the thickness, x is the distance from the bottomset bed start and t0 and X are constants.
Tom Wilson, Department of Geology and Geography

5. 4.7 & 4.10 on due thisThursday.
Part of a continuum where settling time is governed by Stokes law
Tom Wilson, Department of Geology and Geography

6. The problem assignment (see last page of exercise), will be due next week. The exercise requires that you derive a relationship for specific frequency magnitude data to estimate coefficients, and predict the frequency of occurrence of magnitude 6 and greater earthquakes in that area.
Best Fit
Tom Wilson, Department of Geology and Geography

7. Returning to the Gutenberg-Richter Relation
we have the variables m vs N plotted, where N is plotted on an axis that is logarithmically scaled. -b is the slope and a is the intercept.
Tom Wilson, Department of Geology and Geography

8. However, the relationship
indicates that log N will also vary in proportion to the log of the fault surface area. Hence, we could also
There are different perspectives we can bring to bear on the analysis of our data.
Tom Wilson, Department of Geology and Geography

9. Gutenberg Richter relation in Japan
Tom Wilson, Department of Geology and Geography

10. "Best fit" line
In this fitting lab you’ll calculate the slope and intercept for the “best-fit” line
In this example -
Slope = b =-1.16
intercept = 6.06
Tom Wilson, Department of Geology and Geography

11. Recall that once we know the slope and intercept of the Gutenberg-Richter relationship, e.g. As in -
we can estimate the probability or frequency of occurrence of an earthquake with magnitude 7.0 or greater by substituting m=7 in the above equation.
Doing this yields the prediction that in this region of Japan there will be 1 earthquake with magnitude 7 or greater every 115 years.
We can use these relationships predictively. We may not witness a magnitude 7 earthquake, but there is some probability that one could occur
Tom Wilson, Department of Geology and Geography

12. There’s about a one in a hundred chance of having a magnitude 7 or greater earthquake in any given year, but over a 115 year time period the odds are close to 1 that a magnitude 7 earthquake will occur in this area.
Tom Wilson, Department of Geology and Geography

13. In this case, the historical record bears out our prediction
Historical activity in the surrounding area over the past 400 years reveals the presence of 3 earthquakes with magnitude 7 and greater in this region in good agreement with the predictions from the Gutenberg-Richter relation.
Tom Wilson, Department of Geology and Geography

14. Another way to look at this relationship is to say that it states that the number of breaks (N) is inversely proportional to fragment size (r). Power law fragmentation relationships have long been recognized in geologic applications.
Tom Wilson, Department of Geology and Geography

15. Fractal behavior
Tom Wilson, Department of Geology and Geography

16. Relationship described by power laws
Box counting is a method used to determine the fractal dimension. The process begins by dividing an area into a few large boxes or square subdivisions and then counting the number of boxes that contain parts of the pattern. One then decreases the box size and then counts again. The process is repeated for successively smaller and smaller boxes and the results are plotted in a logN vs logr or log of number of boxes on a side as shown above. The slope of that line is the fractal dimension.
Tom Wilson, Department of Geology and Geography

17. Where else does line fitting come in handy? Basic pump test data
Original data showing drawdown during pumping and recovery after pumping ceased.
Recovery phase data after transformation, which includes a log transformation of the observation times.
Tom Wilson, Department of Geology and Geography

18. A pilot carbon sequestration site here in the Appalachians
Tom Wilson, Department of Geology and Geography

19. Tom Wilson, Department of Geology and Geography

20. Residual track relative to the regression line for a horizontal well
Tom Wilson, Department of Geology and Geography

21. Problems 4.7 and 4.10 are due this Thursday.
Some due dates
Problems 4.7 and 4.10 are due this Thursday.
Computer lab – settling velocity problems Parts 4 & 5 of the computer lab are also due this Thursday
Problems 3.10 and 3.11 from the text have already been submitted and returned- so only parts 4 & 5 are left to do.
Tom Wilson, Department of Geology and Geography

22. Let’s get started on today’s lab
Estimating the coefficients of various Mathematical relationships in Geology
Tom Wilson, Department of Geology and Geography