Spatial Variability 2

Why study spatial variability? Soils are spatially heterogeneous at every scale. Soil variability affects every one of our measurements and predictions. But the variability in soils has spatial patterns. Understanding the patterns both gives us more information about what caused them and what processes they might affect, and opens the possibility of using some statistical tools to account for them. For example:

The Analysis of Variance (ANOVA) that we use so frequently in field trials has some built-in assumptions:

• Normally-distributed variables of interest
• All treatments have the same variance
• Pre-existing spatial biases can be eliminated by correct layout of blocks and/or treated as experimental error.

This third assumption is often violated. For example, we might have a soil spatial pattern that has a gradient in one direction, and we block in the other direction. Or (more likely) the variability doesn’t have a simple linear pattern, and there are more than a single gradient present: for example, wind in one direction, drainage in another, and aspect (direction that the slope faces) in a third, so that blocking doesn’t eliminate all of the patterned variability.

One of the points of looking at spatial patterns is that patterned variability can be eliminated in ways that purely random variability cannot. Different approaches for handling this variability include:

• Localize your sampling. First, make sure that each treatment abuts every other treatment in every block. Then, whenever you sample, take a pair of samples for each treatment pair: one (say) in treatment A right next to treatment B, and the other in B right near the A sample. This way, each paired comparison has minimal spatial separation, minimizing the spatial variability effect. Disadvantages: if you have lots of treatments, the number of plots increases quickly. Also, this only works for point samples, and such samples usually have some edge effects. Example: if 2 cultivars differ in water use, sampling on each side of the border will underestimate the difference between them, because water will be moving and so the boundary will be gradual, not sharp.
• Nearest Neighbor Analysis (NNA). This idea was proposed decades ago, starting from the observation that neighboring plots in a variety trial had similar yields regardless of the treatment. In other words, instead of having a statistical model that said

PlotYield = Mean + BlockEffect + TrtEffect + Error,

a spatial model was proposed that said

PlotYield = Mean + TrtEffect + SoilEffect + Error,

where this “soil effect” had a different value in each individual plot, and could be calculated in an iterative manner (simple on a spreadsheet, though difficult back then before computers). This “soil effect” is often larger than the block effect, so you can reduce the error term and get more significance in your experiment. In fact, sometimes too much significance…

· Using semi-variograms and other tools of spatial statistics, you can calculate the spatial patterns and eliminate them in a pre-planned, less after-the-fact manner than with NNA. I won’t cover these tools in this class, except to say that if you think you need it in your research, read about it before you plant.

• Miller & Miller scaling, also known as similitude scaling (pages 664-666 in Hillel). This idea is very elegant and powerful, though it is clearly based upon unrealistic assumptions. Suppose that, across a landscape, the soils vary only in terms of the mean particle size. That is, by sampling 2 soils, one would be identical to the other if you changed the level of magnification at which you were viewing it. This means that each soil has a characteristic length, say Lambda. Now if you choose one soil as a reference, the other soils will differ from the reference by some ratio alpha_i = lambda_i / lambda_ref. Now consider that:

Capillary rise differences are a function of pore radius.

Permeability (k, not K) is a function of pore radius squared.

This suggests that, by knowing lots of properties of the reference soil, and knowing the alpha_i for every other soil, you could estimate all of their properties. This approach frequently works.

Characterization of spatial variability and spatial patterns is still in its infancy. Consider that nothing we’ve seen says whether two similar regions are connected. But this is frequently critical. For example, knowing the semi-variogram of a sandy soil within a landscape does not tell you whether that soil occurs in isolate patches, or as a continuous feature (such as a river). This is analogous to our earlier discussions of soil structure, where we saw that the key to having high hydraulic conductivity was to have the big pores interconnect.

A new spatial concept, a connectivity index, was recently developed by Western (Melbourne, Australia) and Bloschl (Vienna, Austria). They pointed out that two figures, say A and B below, may have identical semi-variograms but completely different connectivities. Because connection is critical to flow, they developed a connectivity function rather like a semi-variogram to characterize this concept.

As a real test of this concept, Western and Bloschl analyzed spatial patterns of soil wetness in a watershed at various overall wetness levels. They found that, at low to intermediate wetness levels, there was a connected structure of wetter soils – so connectivity of the structure was a function of wetness. Because as a rule wetter soils have higher hydraulic conductivity, this further suggests that the soil has evolved drainage patterns that emerge only at specific wetness levels. They also demonstrated that the degree of connectivity shown in the field was much greater than would occur in a random pattern.