## Interpolation Methods and Model Prediction

Sampling measurements made at discrete points, such as measurements of contaminant concentrations, can be used to build a model for the whole site. Different methods are available to make models for contaminant concentrations at all points within the site. Simple, more complex, and advanced interpolation methods can support these models, provided that appropriate data requirements are met.

#### Categories of Geospatial Interpolation Methods

The major geospatial methods can be grouped into three categories depending on their complexity: simple, more complex, and advanced. These methods are presented in Table 3.

**Table 3. Organizing geospatial interpolation methods**

Simple | More Complex | Advanced | |
---|---|---|---|

Description | Mechanistic methods with no statistical error model | Regression methods with no spatial correlation model. | Extension of regression methods including a spatial correlation model. Also known as geostatistical methods. |

Data Requirements | Works best with larger data sets | Works best when other predictor variables are available. | Works best when sufficient data are available to estimate correlation model. |

Statistical Assumptions | None | Regression residuals are spatially uncorrelated and normally distributed (after transformation). | Residuals after trend removal are stationary and normally distributed (after transformation). |

Provide Prediction Uncertainty | None | Yes. Prediction standard error or variance provides measure of uncertainty. True uncertainty is higher when the residuals are spatially correlated. | Yes. Prediction standard error or variance provides measure of uncertainty. True uncertainty is higher because standard error does not include uncertainty from estimation of model parameters. |

Example Methods | Inverse distance weighting (IDW), natural neighbors, Thiessen polygons |
Parametric and nonparametric regression; local linear regression; splines and kernel methods |
Kriging, conditional simulation* |

*Conditional simulation provides a set of possible interpolated values that can be used to estimate the probability distribution of predictions at each location |

Simple methods provide a quick, simple approach to modeling spatial data. First, simple methods are conceptually simpler because they require almost no assumptions be made about the data or the variable to be mapped, only that the sampled data relate to one another in space or time, or both. Second, these methods are computationally simpler, so that large data sets can be efficiently mapped. Because the simple methods do not impose much structure on the data, they generally require more data in order to produce a good interpolation. Third, these methods are statistically simpler because they do not include a statistical error component that can be used to estimate the error in the predictions at unsampled locations.

In contrast, more complex and advanced methods make assumptions about the statistical distribution of a sampled population and can provide estimates of prediction error. This error derives from the inability to sample everywhere, and from the inability of the method to match the data. More complex methods include regression type methods, which are based on statistical assumptions about the data but do not include an explicit representation of spatial correlation. See GSMC-1, Section 5.5.1 for information about linear regression. The advanced methods, including kriging and conditional simulation, are based on more stringent statistical assumptions and include an explicit model of spatial correlation. The advanced methods are also commonly known as geostatistical methods.

The output of every geospatial interpolation method is the set of interpolated values for unsampled locations of interest. The inputs to the method are the observed data. The method can be thought of as a mathematical equation that converts the sampled inputs into the interpolated outputs. This equation or model of the data will be complicated in some cases, but the mathematical details are handled by the software and do not need to be fully understood by the user. One way of thinking about how the geospatial interpolation methods differ is to distinguish between the different components (spatial trend, spatial correlation, error) that are represented within the methods. The different components for the categories of methods are illustrated here:

Simple Method | = spatial trend component |

More Complex Method | = spatial trend component + (secondary data component) + error component |

Advanced Method | = spatial trend component + spatial correlation component + error |

*Simple Methods* ▼*Read more*

*More Complex Methods* ▼*Read more*

*Advanced Methods* ▼*Read more*

*Measures of Uncertainty* ▼*Read more*