Marina Marcelli

Question:

What is the spatial distribution of the water table in my study area with respect to scale? How are wells in my area clustered? Is there a relationship between wells where the water lies above the first lava and below the first lava, wells that have water tables that correspond to the first lava those that lie above the water table, and wells that have water tables that correspond to the water table and those that lie below it?

Approach:

I used both the Ripley’s K function and Ripley’s Cross K function to look at the spatial distribution of water, first lava and the water table with respect to lava in the area. Like the variogram and the cross variogram, Ripley’s K function and the Cross Ripley’s K function describe spatial distributions at different scales. However, rather than use the variance, as with the variogram, Cross Ripley K compares the spatial data to a curve that represents complete randomness (the Poisson’s curve).

Brief Methodology:

I first did a preliminary analysis using the Ripley’s K function for both the depth to first water and the depth to first lava.

Then I used a Kcross to compare the differences between the depths to first lava and water. Because the Kcross function uses only factor variables, I had to make sure my data was categorical. I thus decided to bin my data into three categories depending on the Lava – Water (L-W) value.

L_W[i] >40 <- “above”

40>= L_W[i] >= -40 <- “equal”

L_W[i] <-40 <- “below”

Where “above” stipulates that the water table is above the contact to first lava, “equal” stipulates that the water table roughly equated to the contact to first lava, and “below” signifies that the water was below the contact to first water.

After binning the data I compared each category to the others, resulting in three Cross Ripley’s K plots. I then plotted a significance envelope to see where the data was actually significant.

Results:

Figure 1: Study area with the well logs used for this study. The cyan represent the well logs that have a water table “above” the first lava. The Blue are below, and the pink have a water table that roughly correponds to the first lava.

Figure 2: K(r) vs r, the distances for which we are comparing clustering. For these data, Poisson’s curve appears to be nearly horizontal. This means that the data appear to be clustered at all scales measured by Ripley’s K function. According to these plots the depth to first lava data is clustered at all scaled.

Figure 3: The shape of the depth to first water Ripley’s K plot is different than the depth to first lava plot (fig 2). What that means, I don’t know. However, based on both Poisson’s curve and the significance envelope, water also clusters all scales measured.

Figure 4: Ripley’s Cross K function for the points where the water table is above the contact with first lava, with the points where the water table is below the contact with first lava. At distances shorter than 6 km, the spread appears to be random, while distances, the data appear to have significance. This means that the data do not cluster a close distance. This corresponds what we would expect from natural fluctuations in elevation of the water table driven by changes in elevation. In a simple system, with one lava and one water table, this works well. However, the study area is in reality much more complicated than this.

Figure 5: Cross K function for the points that correspond to above and equal. The data appear to be correlated at much closer distances than the above and below data.

Figure 6: Cross K plot for points corresponding to equal and below. They appear to be linked at all scales. Ideally they would be clustered at closer scales and random farther away. The discrepancy might be accounted for by faults, or multiple water bearing layers.

Conclusion:

Depth to first lava and depth to first water are linked at all scales measured by Ripley’s K plot. In this case, the largest scale I managed to measure was 12 km. Wells that plotted as above the water table and wells that plotted below the water table were not clustered, rather they showed to be linked at distances greater than 6 km. Points that were equal and below  were correlated at all scales. Well logs that were equal and above were correlated at scales larger than 2 km. This might have to do with lack of data. It might also have to do with the regional geology.

Critique of the method:

One aspect of the process that I walked away from was that my field area is 40 km across. The largest r value I calculated was 12 km. Ripley’s K plots and the Cross K plots might demonstrate different relationships are larger scales. In the future I would like to figure out how to change the r values.

I will need to be able to plot these data at larger scales to determine weather or not they corroborate what the variograms found.