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In the bivariate formulation, interpolation requires the fitted function to match the measured values exactly at every given data point, expressed as $z_i = F(x_i, y_i)$. Approximation relaxes this exact fitting by allowing a deviation or error term $e(x_i, y_i)$, expressed as $z_i = F(x_i, y_i) + e(x_i, y_i)$.

Conditions: Given m points $(x_i, y_i, z_i)$ with $i=1,m$.; Find a function $F(x,y)$.; Interpolation requires exact matching at the sample points.; Approximation allows an error term $e(x_i, y_i)$.

In the bivariate formulation, interpolation requires the fitted function to match the measured values exactly at every given data point, expressed as $z_i = F(x_i, y_i)$. Approximation relaxes this exact fitting by allowing a deviation or error term $e(x_i, y_i)$, expressed as $z_i = F(x_i, y_i) + e(x_i, y_i)$.

Conditions: Given m points $(x_i, y_i, z_i)$ with $i=1,m$.; Find a function $F(x,y)$.; Interpolation requires exact matching at the sample points.; Approximation allows an error term $e(x_i, y_i)$.

The groundwater example requires quadrivariate interpolation because the observations are scattered in three-dimensional space and also vary over time. This involves four variables: three spatial dimensions and time.

Conditions: Input values are measured at 3D points over time.; Output is a 4D raster combining 3D space and time.

The groundwater example requires quadrivariate interpolation because the observations are scattered in three-dimensional space and also vary over time. This involves four variables: three spatial dimensions and time.

Conditions: Input values are measured at 3D points over time.; Output is a 4D raster combining 3D space and time.

Bivariate interpolation is the process of finding a surface representation based on values measured at 2D points. Its main applications include converting scattered points, profiles, or isolines into a raster format, computing a denser or smoother Triangulated Irregular Network (TIN), and resampling continuous-data rasters, such as elevation, to a higher resolution.

Conditions: Input values are measured at 2D points.; Output is a surface representation, raster, or denser/smoothed TIN.

Bivariate interpolation is the process of finding a surface representation based on values measured at 2D points. Its main applications include converting scattered points, profiles, or isolines into a raster format, computing a denser or smoother Triangulated Irregular Network (TIN), and resampling continuous-data rasters, such as elevation, to a higher resolution.

Conditions: Input values are measured at 2D points.; Output is a surface representation, raster, or denser/smoothed TIN.

In the “surface points in time” visualization, the vertical axis represents the distribution in time, not the actual three-dimensional physical location or depth. Each point still has an x, y horizontal location, but the plotted vertical extent encodes elapsed time, as illustrated by lengths corresponding to 250 days and around 500 days.

Conditions: The visualization is the left panel labeled “surface points in time”.; The data represents groundwater measurements over time.

In the “surface points in time” visualization, the vertical axis represents the distribution in time, not the actual three-dimensional physical location or depth. Each point still has an x, y horizontal location, but the plotted vertical extent encodes elapsed time, as illustrated by lengths corresponding to 250 days and around 500 days.

Conditions: The visualization is the left panel labeled “surface points in time”.; The data represents groundwater measurements over time.