Interpolation Deviations & Sampling Density

After the optimal interpolation method parameters were identified, the split-sample process was repeated 200 times for each sampling density of original measurements (See Ch. 3 Methodology). As expected, the deviations of all three interpolation methods decreased when using more depth measurements (Figure 16). Many studies use a statistic such as RMSE as a global indicator of accuracy (Aguilar et al., 2005). However, this measure will be inflated in areas of deep water. Thus, the absolute median percent deviation was used instead, as it would not be as sensitive to the depth of water, and would allow for more accurate comparisons and applications to other study sites in deeper or shallower waters.

Figure 16. Absolute median percent deviation as a function of sampling density.

The largest differences in absolute median percent deviation between the three interpolation methods occurred when using 1% of the original measurements. As the percentage

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of data used for interpolation increased, the median distance to the nearest measurement decreased, and the accuracy of all methods became more similar. Since the most notable differences between the interpolation methods occurred when using 1% of the original measurements, further analysis focused on this sampling density in order to determine the strengths and weaknesses of the various interpolation methods in relationship to terrain characteristics.

The differences between the three interpolation methods were very small when using 50% of the original measurements because there were typically measurements 1 cell away from a cell to be interpolated. Since the accuracy is comparable between the three methods at this sampling density, the choice of interpolation method should then be chosen based on computational and time resources. IDW is the fastest and least computationally intensive method and would therefore be adequate for most purposes when the median distance to measurements is approximately 1 cell. This finding supports previous studies that have indicated that IDW is comparable to much more complex statistical methods, such as kriging, when using higher sampling densities (Lloyd and Atkinson, 2002; Ali, 2004; Blaschke, 2004; Podobnikar, 2005; Chaplot et al., 2006; Liu et al., 2007). From Figure 16, the optimal sampling density can be identified for each interpolation method for this study area, depending on your required criteria of accuracy. The absolute median percent deviation for each method as a function of sampling density can be used to estimate interpolation uncertainty with a decaying power function (Table 4).

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Table 4. Equations to estimate interpolation uncertainty using the sampling density. Y represents the interpolation median percent deviation and X represents the percent sampling density.

Interpolation Method Equation R2

IDW y = 0.7676x-0.524 _0.9997

Spline y = 0.2236x-0.512 _0.9537

TIN y = 0.3502x-0.553 _0.9866

However, the equations in Table 4 provide global estimates of uncertainty and assume the median distance to the nearest measurement (Table 5) for every cell, which typically is not the case with sparse depth measurements. The inadequacy of these equations is addressed by deriving equations to estimate interpolation uncertainty using distance to the nearest measurement in Section 4.7. The median percent deviation had the greatest decrease when the sampling density increased from 1% to 5%. This is likely because the median distance to the nearest measurement decreases from 5 cells to 2 cells with this 5-fold increase in sampling density. When the sampling density increases 5-fold from 5% to 25%, the median distance to the nearest measurement only decreases from 22.26 meters (2 cells in this study) to 10 meters (1 cell in this study; Table 5). This smaller decrease in the median distance to the nearest measurement corresponds with a similar smaller decrease in absolute median percent deviation in Figure 16 compared to the changes in deviation between the 1% and 5% sampling densities.

Table 5. Relationship between sampling density and median distance to the nearest measurement.

Sampling

Density (%) Nearest Measurement (m) Median Distance to the Nearest Measurement (cells) Median Distance to the

1 50 5

5 22.26 ~2

10 14.14 ~1

25 10 1

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4.3 Correlations

The split-sample and raster query processes were repeated 200 times at each sampling density with the optimal interpolation parameters. The distance to the nearest measurement, slope, and curvature rasters were queried for every interpolation percent deviation and the Spearman’s correlations were calculated between the percent deviation and these explanatory variables. The positive correlation between the absolute percent deviation and distance to the nearest measurement shows a similar decreasing trend with higher sampling densities for all methods (Figure 17). This decreasing trend suggests that the distance to the nearest measurement is a better predictor of interpolation accuracy at low sampling densities, while at higher sampling densities, all unconstrained cells will be close enough to cells constrained by measurements that the interpolation deviation will be more related to the terrain, and especially curvature.

Figure 17. Spearman’s Correlation Rho between absolute percent deviation and distance to the nearest measurement as a function of sampling density.

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Absolute percent deviation and slope are also positively correlated with each other for all methods. However, IDW has a much stronger positive correlation with slope than TIN and spline (Figure 18). TIN and spline show similar decreasing trends with higher sampling densities, while IDW remains relatively constant at all sampling densities. IDW is most strongly correlated with slope as a result of the weighted-average algorithm, which can cause the interpolated depth to be much different than surrounding measured depths in areas of steep slopes.

Figure 18. Spearman’s Correlation Rho between absolute percent deviation and slope as a function of sampling density.

The correlation between absolute percent deviation and absolute curvature increases with greater sampling densities for all methods, but the correlations also indicate differences in the interpolation methods. TIN interpolation is the most strongly correlated with absolute curvature, followed by spline, and IDW is the least correlated with curvature (Figure 19).

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Figure 19. Spearman’s Correlation Rho between absolute percent deviation and absolute curvature as a function of sampling density.

When the signs of percent deviation and curvature are taken into account, the correlation shows the same trends as the absolute percent deviation and absolute curvature, but the correlation is greater for all methods (Figure 20). This reflects the relationship between whether the interpolated depths are deeper or shallower than the measured depths and the surface convexity or concavity of the measured depths.

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Figure 20. Spearman’s Correlation Rho between percent deviation and curvature as a function of sampling density.

A simple 2-D cartoon profile illustrates the relationship between TIN interpolation deviations and curvature, and how the sign of the interpolation deviation changes with convex and concave surfaces (Figure 21). Further analysis on the relationship between interpolation deviations and terrain characteristics is provided in the next two sections.

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Figure 21. 2-D cartoon profile showing relationship between the sign of TIN interpolation deviations and the surface curvature. TINs are typically deeper than the measured surface when the measured surface is convex and TINs are shallower than the measured surface when the surface is concave. TINs tend to become more accurate where the curvature of the surface is zero at the inflection point.