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

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

3. Active faults of Japan

5. Seismic profile of accretionary prism at Nankai Trough

7. This problem assignment (see last page of exercise), will be due next Tuesday. The exercise requires you to 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.

8. I. II. III. Gutenberg-Richter Frequency Magnitude Relationship
Seismic moment - M
where r is the characteristic linear dimension of the fault
II.
Moment-magnitude relationship
III.
where c is often 1.5

9. Gutenberg Richter relation in Japan

10. "Best fit" line
In today’s lab we will calculate the best-fit line and calculate the slope and intercept for this line
In this example -
Slope = b =-1.16
intercept = 6.06

11. 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.

12. 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.

13. The relationship we derived above between the frequency of earthquake occurrence (N) and the characteristic linear dimension (r) of the fault surface
are referred to as power laws. In general form a power law is written as
We could rewrite the above example from seismicity as

14. Box counting is a method used to determine the fractal dimension
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.

15. * Note that Q in this expression is the pumping rate.
Logarithmic relationships are also common in hydrological applications. For example pump test measurements of drawdown (s) over time (t) are related to aquifer transmissivity by the following equation
* Note that Q in this expression is the pumping rate.
The hydraulic conductivity K is derived from transmissivity and then K is used to estimate the maximum sustained yield (Q) of the aquifer, where Q is

16. 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.