Sachin MehtaUniversity of Nevada, Reno

Point Interpolation

Overview

The general purpose of this report is to discuss the interpolation a set of points to create four predicted surfaces. The points that were used represent pollution samples taken along the Maas River measured in parts per million (ppm). The four surfaces will be created in Arc Map using the tools found in the Geo-statistical Analyst. The created surfaces will then be used to predict the occurrence of a specified pollutant along the flood plain of the Maas River. For this exercise I chose to look at the spatial variation of Mercury along the flood plain of the Maas River.

The tools I chose to use prior to the interpolation were the histogram (Fig. 1) to represent the distribution of the chosen data and the semi-variogram cloud (Fig. 2) to look at the spatial autocorrelation of the data. For the histogram a logarithmic transformation was used to normalize the data, which was skewed to the left. The semi-variogram for Mercury indicated that the samples only varied slightly in relation to distance. For the actual interpolation I chose to use an Inverse Distance Weighted (IDW), two different Local Polynomial Interpolations (LPI), and one Kriged surface. Each tool brings with it some advantages and some disadvantages…the created surfaces and subsequent results would then be compared to determine the best fit.

Fig. 1: Histogram for Mercury

Sachin MehtaUniversity of Nevada, Reno

Point Interpolation

Fig. 2: Semi-variogram for Mercury.

Methodology

The first method chosen was the Inverse Distance Weighted (IDW) because it is known to be an exact interpolator. The idea behind the IDW is loosely based on Tobler’s law that states: “that things closer together are more closely related than those that are farther apart.” The IDW has a limited number of variables that can be changed, such as the power function and the neighborhood type. So with that in mind I chose to leave the power function at 2 and the neighborhood type to smooth. The smooth feature creates multiple ellipses to be used in the interpolation, while the power function is used to justify the best fit.

The second method that was used was the Local Polynomial Interpolation (LPI). The idea behind LPI is the use of varying orders of polynomials to interpolate short-range variation within a search area or neighborhood to produce an output surface. In the first LPI there was only one variable that was changed was the kernel function which was changed from a polynomial function to Gaussian. The Gaussian function is a bell-shaped function it seems to normalize the data slightly. For the second LPI everything remained the same as the first LPI except for the use of the optimize button, which minimizes the mean square error.

The last method used in this analysis was a spatial analysis tool named kriging. Kriging uses the concept of auto-correlation which tend to provide a better measure of certainty or accuracy than the other method which use deterministic interpolation. The first option that was change for the analysis was the kriging type, it was changed from a simple to an ordinary. The ordinary kriging is an examination of the trend, if it is unknown and is composed of a linear function it will fit, if it varies or is known one of the other types of kriging will be used. The next step was to use a first order trend removal, this was done to provide a normality or stationarity (special

Sachin MehtaUniversity of Nevada, Reno

Point Interpolation

homogeneity). There were two steps that followed first was to change that kernel function to Gaussian, then use the optimize button. The Gaussian again normalizes the data and the optimize button minimizes the mean square error. The neighborhood type was left as standard.

Results

The results varied widely with some of the created surfaces over-predicting while others under-predicted these can be found in (Table 1). Analyzing the IDW created surface found in (Fig. 3) it is noticeable that it created small concentric circles around some points, but not around all the points. This could be attributed to the fact that the IDW is an exact interpolator and the values of the surrounding points weren’t the same. The predicted values didn’t show much change from the actual values which could be a good indicator that is this surface might be really close to what might actually be occurring along the Maas River.

Fig. 3: Inverse Distance Weighting created surface of Mercury

Sachin MehtaUniversity of Nevada, Reno

Point Interpolation

The IDW had a Root Mean Squared (RMS) value of 18.69326 which is relatively high. This could indicate that the IDW created surface is grossly over-estimating, though is doesn’t actually seem like it after a visual comparison. The next method used was the LPI, the results can be found in (Fig. 4) and (Fig. 5). The first surface that was created had two portions that were completely blank, one on the bottom and the other in the right hand corner. This could be attributed to the fact that nearest points were too far away for the LPI to create a predicted surface. The RMS value for this surface was 14.36267 which is relatively high. This could be found in the surface values, which slightly under-estimate at the low end and grossly over-estimate on the high end when compared to the actual sample values.

The other surface that was created didn’t have such large portions that were left blank, but they were still there. This is because there were no points close enough for the LPI to predict what values should cover that particular surface. This surface had a slightly higher RMS value of 16.1452, but a low Root Mean-Square Standardized (RMSS) value of 0.97938. This would indicate that the surface created is an underestimate, this noticeable when comparing the created values with the actual values. The lower values of the predicted surface were far too low and the higher values were also lower than they should have been.

Fig. 4: Mercury LPI Analysis Fig. 5: Mercury LPI (Optimized) Analysis

Sachin MehtaUniversity of Nevada, Reno

Point Interpolation

The last surface that was created was the Kriged surface which is found in (Fig. 6). The first noticeable difference is the large portion in the lower right that is blank. This is a result of the neighborhood search area being too small, so the resulting surface tends to exclude some areas. In this case the area which has no almost no data points, which is very similar to the non-optimized LPI created surface. Something else that stands out is how jagged the surface is compared to the other surfaces which look relatively smooth in comparison. This could be because the Kriged surface tends to create a better measure of certainty or accuracy.

Fig. 6: Kriged Surface for Mercury

The Kriged surface produced a RMS of 15.32313 and a RMSS of 1.18722, which are good indicators of slight over-estimating. This can be found when comparing the Mercury samples to the values produced by the Kriged surface. The lower end seems to be only slightly over estimated, while the upper-end results are just a little higher only at the extreme. This would conclude the results portion of the analysis.

Table 1: Results of Analyses for Mercury

MethodMin_value

(ppm)Max_value

(ppm)Percent area

>20 ppmPercent area

>60 ppmPercent area

>160 ppm          

         IDW 14.02470303 127.5209198 3 89.9 7.1LPI 10.49144936 158.2221069 12.2 76.9 10.9

LPI (optimized) -0.534780145 116.6180954 0 0 0Kriging 9.593212128 152.0141754 13.7 76.2 10.1

         

Sachin MehtaUniversity of Nevada, Reno

Point Interpolation

Conclusion

This analysis produced some interesting results and not being able to get percent area for the LPI (optimized) through off the comparison a bit. In my opinion the kriged surface produced the optimal results. Though the IDW seemed to be the most accurate, they did not appear to be realistic. Then again none of these results seemed very realistic because other variables such as elevation, course of the river, distance to river, and other heavy metals were not compared. If elevation was considered it could be used to determine if there were any hills or sinks where the presence of the Mercury might be higher or lower. The course of the river and distance to river play an important part because it could take a turn at the top or the bottom which could indicate why some Mercury sample are found in certain areas. The last uncertainty would be to compare the other heavy metals that were taken and determine if they accumulated in the same areas. Without comparing these other variables the analysis seems to come up a little short of expectations.