Inverse Distance Weighting (IDW) is an exact spatial interpolation method that uses a function of the inverse difference between the estimate location and the observed locations. The IDW method assumes that the closer the observed values are to the value being estimated, the more influence they have on the estimated value. This follows Tobler's first law of geography, "everything is related to everything else, but near things are more related than distant things". This method tends to preserve minimum and maximum values.

Spline is another exact spatial interpolation method that uses a polynomial equation to estimate values from observed values. This function minimizes the overall surface curvature while passing through the observed values. In turn, minimum and maximum values may not be preserved in this method. However, the surface created is a smooth surface. The smoothness of the function can being directly attributed to the number of sample points, with a large number of sample points creating a smoother surface.

Kriging is a spatial interpolation method that relies on autocorrelation. It is similar to IDW, where the closer the observed values are to the value being estimated, the more influence they have on the estimated value. However, it differs where Kriging fits a mathematical function to the points using an exploratory method. The process will go through many different functions before it selects the best fitting function that represents the data well.

I used 1,234 training points and 50 test points for each of the analyses run. Also for each analysis, I used a cell size of 1,000 meters. This created a sharper raster dataset output. For IDW, I used the default power of 2, as when I changed the power to 1 or 3, the output was too smooth (overfitting) or it was not as smooth. I also used the variable search radius and the default 12 number of points.

 For the Spline interpolation method, I used a Tension spline type with a weight of 0.25 and 12 points. The Regularized spline type was creating an output with not many cells with middle range values, as it was assigning < 0.1" or > 1" for most of the output. The tension spline type fixed this issue a little bit as well as adjusting the weight to 0.25 from the default of 0.1. For the Kriging method, the only parameter I changed was the semi-variogram model from spherical to Gaussian. This created an output that represented the data better.

The original dataset had a minimum value of 0, with a maximum value of 7.16. The mean of the dataset was 0.3944 with a standard deviation of 0.4914. The IDW method predicted the minimum and maximum values very close, with 0 and 7.0892 respectively, as we expected. The mean of the IDW dataset was the highest of the three methods at 0.4416, although it was the 2nd closest to the actual value. The standard deviation was noticeably smaller than the original dataset, at 0.2244. We can see from the test values that the IDW method was very good at approximating the observed precipitation values. In the comparison of the symbolized output dataset between the 3 methods, we see a very smooth image that represents the dataset well.

The Spline method predicted the a negative minimum value and a maximum value far from the original value, with -2.1618 and 2.9144 respectively. The mean of the Spline dataset was the lowest of the three methods at 0.3208, and it was the farthest to the actual value of the three means. The standard deviation was larger than the original dataset, at 0.8287. Like the IDW method, we see from the test values that the Spline method was very good at approximating the observed precipitation values. However, it did predict some negative values, which is impossible for precipitation values. Also, in the comparison of the symbolized output dataset between the 3 methods, we see some sharp increases and decreases that make the image rough. In some areas, we see a jump from < 0.1" to > 1" over a short distance.

The Kriging method predicted a close to 0 minimum value, at 0.098, but the maximum value was only 0.8398. The mean of the Kriging dataset was the closest to the actual mean of the three methods at 0.436. The standard deviation was smaller than the original dataset, at 0.8287, and the smallest of the 3 methods.  We can attribute this to the small range of values that the method produced. We can see from the test values that the Kriging method did a very poor job at approximating the observed precipitation values. This method produced wildly inconsistent estimated values. In the comparison of the symbolized output dataset between the 3 methods, we see a smoother image than the Spline method, however, the full range of values were not included in the data. In the areas where we would expect to see > 1" of precipitation, we find that the estimated values are < 1".