Geology 351 - Geomath Estimating the coefficients of linear, exponential, polynomial, logarithmic, and power law expressions tom.h.wilson tom.wilson@mail.wvu.edu Department of Geology and Geography West Virginia University Morgantown, WV Tom Wilson, Department of Geology and Geography

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

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

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

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

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

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

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

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

"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

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

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

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

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

Fractal behavior Tom Wilson, Department of Geology and Geography

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

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

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

Tom Wilson, Department of Geology and Geography

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

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

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