Question:

How does the variance between the depths to the first lava flow in my filed area vary with increasing distance?

The Tool: Variogram

I used a variogram to analyze the variance within my dataset. Variograms are discrete functions calculated using to measure the average correlation between pairs of measurements at various distance (Cameron and Hunter 2002). These distances are referred to as the binned distances (Anselin, 2016). In this study, binned distances determine the distances by which the depth to first lava flow is autocorrelated (Aneslin, 2016).

variog(geodata,coords = geodata$coords,data = geodata$data,max.dist=”number”) (R documentation)

The R code above need the geodata, an array of the data you are testing, the cords, or the coordinates those data correspond too.

Brief Methodology:

I selected data based on the quality of the descriptions, in the well log and assigned each well log I have interacted with a value from 0 to 3. Data with a score 0 represents either a dry well or a destroyed well. Data that has a score of 3 is well described and denotes either the depth to first water or the standing water level. I used data with a score of 3 for this analysis. I denoted the depth to the first lava flow in each of the well logs.

Figure 1: My field area, the well log locations are the blue circles, a rough delineation of my field area is in while.

First I normalized my data, giving a mean of 0 and a standard deviation of 1. Then I determined a max distribution, which determines the max bin size the variogram uses. The max distribution determines the lag increment, or the distances between which the increment is calculated (Anselin, 2016).

The data are projected, meaning that their horizontal distance is measured in meters. However, they are recorded in decimal degrees and located at the center of the township and range section that they are located in. Rather than convert my data from decimal degrees to meters, I recognized that there are around 111 km in 1 degree (at the equator), that there are 0.6214 miles in a kilometer and that there are 6 miles in a township. This helped me determine the max distribution for the variog function in R. I decided upon a max distributions of 0.09 and 0.5 degrees.
I then used R’s variog function on the normalized data with the max distribution of 0.09 and 0.5 degrees.

Succinct Results:

Figure 2: Variogram of the MLV-LP-BVM triangle with a max distribution of 0.09 degrees. Note the low semivariance with the lower bin size (.02 to 0.08 degrees).

Figure 3: Variogram of the MLV-LP-BVM triangle with a max distribution of 0.5 degrees. Note the low saw-toothed pattern of semivariance. From 0.01 to 0.08, semivariance is low, it spikes up, and lowers again around 0.2, spikes again at 0.3 degrees and lowers again at 0.4.

Critique:

I tested max bin sizes of 0.5 and 0.09 degrees to see how the variogram changes with an increasing bin size. Changing the max bin size, called the max distribution in R, changes the variogram. The smaller bin size, 0.09 degrees, limits the max bin size to a narrow range of values. In effect, the variogram only tests the covariance of data points that are separated by a maximum .09 or degrees. Increasing the bin size increases the distance around which the covariance of the data points are tested. Thus, mad distributions of 0.5 result in a spikey plot. Normally, one would expect the variogram to plateau at larger bin sizes, representing large variance with the data with larger distances, but figure 2 does not.

At its simplest, the variogram in figures 1 implies that data points that are correlated in space are more likely to have similar depths to the first lava flow. You can see this in the low variance in you find in locations that are close to each other, the smaller distances, and the higher variance in distances that are farther from each other.

Figure 2, with the max distribution of 0.5 degrees one could made an argument for a distribution of the locations of lava flows based on the locations of volcanic centers. Medicine Lake Volcano lies in the North of the study area and Lassen Peak lies in the south. The two spikes in variance with location might be linked to the distances between those two centers. In other words, distances that correspond to low values of semivariance (>1) correspond to either regions that lie on the same lava flow, or another near surface lava flow sourced from another volcanic center in the region (figure 3). Rather than finding the same lava flow at depth, we are seeing different lava flows at similar depths.

Figure 4: My field area with the rough delineations of two of the near surface lava flows in the region. 1 degree of longitude corresponds to roughly 111 km.

Lava flows are not attracted or repulsed to or from each other, but they do follow the laws of physics. Often, Volcanoes build topography, when lava erupts from them, the lava will flow from the high elevations of the volcano, to basins, using paleo-valleys as conduits for flow. Thus, if you know the paleo topography you can understand where a lava might have flowed, and where it might emplace. Predicting paleo-topography can be difficult in old volcanic regimes, but on the geologic timescales we are looking at, I can predict that the topography of the MLV-LP-BVM triangle has not changed much over the past 5 ma.

Lavas flows from high to low topography and from Volcanos. The two main volcanos in my field area are Lassen Peak in the South and Medicine Lake Volcano, in the North. Moreover, lava flows from high to low elevation. Lavas emplace in basin, if they sit long enough, the top soils form, the basin subsides, and another the cycle repeats.

My data does have different spatial patterns at different scales. If you look at regions that are within 0.1 of a degree of distance then you would expect to see a similar depth to the lava flow. If you move to 0.5 of a degree, you see a sees-saw effect, where the depth to the lava flow moves from having lava close to the surface of deeper down. This variation stems from the proximity of the data point to the sources and the sinks of the lava flow.

I would use this technique again, it helped me think about my problem.

Sources:
Cameron, K, and P Hunter. 2002. Using Spatial Models and Kriging Techniques to Optimize Long-Term Ground-Water Monitoring Networks: A Case Study. Environmetrics 13:629-59.

Anselin, Luc. “Variogram”. Copyright 2016. Powerpoint File.

R Core Team (2013). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. URL http://www.R-project.org/.