[转贴]：Inverse distance to power

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Inverse distance to Power
The Inverse Distance to a Power gridding method is a weighted average interpolator, and can be either an exact or a smoothing interpolator. With Inverse Distance to a Power, data are weighted during interpolation such that the influence of one point relative to another declines with distance from the grid node. Weighting is assigned to data through the use of a weighting power that controls how the weighting factors drop off as distance from a grid node increases. The greater the weighting power, the less effect points far from the grid node have during interpolation. As the power increases, the grid node value approaches the value of the nearest point. For a smaller power, the weights are more evenly distributed among the neighboring data points. Normally, Inverse Distance to a Power be haves as an exact interpolator. When calculating a grid node, the weights assigned to the data points are fractions, and the sum of all the weights are equal to 1.0. When a particular observation is coincident with a grid node, the distance between that observation and the grid node is 0.0, and that observation is given a weight of 1.0, while all other observations are given weights of 0.0. Thus, the grid node is assigned the value of the coincident observation. The Smoothing parameter is a mechanism for buffering this behavior. When you assign a non-zero Smoothing parameter, no point is given an overwhelming weight so that no point is given a weighting factor equal to 1.0.
One of the characteristics of Inverse Distance to a Power is the generation of "bull's-eyes" surrounding the position of observations within the gridded area. You can assign a smoothing parameter during Inverse Distance to a Power to reduce the "bull's-eye" effect by smoothing the interpolated grid.
Inverse Distance to a Power is a very fast method for gridding. With less than 500 points, you can use the All Data search type and gridding proceeds rapidly.


Kriging
Kriging is a geostatistical gridding method that has proven useful and popular in many fields. This method produces visually appealing maps from irregularly spaced data. Kriging attempts to express trends suggested in your data, so that, for example, high points might be connected along a ridge rather than isolated by bull's-eye type contours. Kriging is a very flexible gridding method. You can accept the Kriging defaults to produce an accurate grid of your data, or Kriging can be custom-fit to a data set by specifying the appropriate variogram model. Within Surfer, Kriging can be either an exact or a smoothing interpolator depending on the user-specified parameters. It incorporates anisotropy and underlying trends in an efficient and natural manner.
The kriging algorithm incorporates four essential details:
1. When computing the interpolation weights, the algorithm considers the spacing between the point to be interpolated and the data locations. The algorithm considers the inter-data spacings as well. This allows for declustering.
2. When computing the interpolation weights, the algorithm considers the inherent length scale of the data. For example, the topography in Kansas varies much more slowly in space than does the topography in central Colorado. Consider two observed elevations separated by five miles. In Kansas it would be reasonable to assume a linear variation between these two observations, while in the Colorado Rockies such an assumed linear variation would be unrealistic. The algorithm adjusts the interpolation weights accordingly.
3. When computing the interpolation weights, the algorithm considers the inherent trustworthiness of the data. If the data measurements are exceedingly precise and accurate, the interpolated surface goes through each and every observed value. If the data measurements are suspect, the interpolated surface may not go through an observed value, especially if a particular value is in stark disagreement with neighboring observed values. This is an issue of data repeatability.
4. Natural phenomena are created by physical processes. Often these physical processes have preferred orientations. For example, at the mouth of a river the coarse material settles out fastest, while the finer material takes longer to settle. Thus, the closer one is to the shoreline the coarser the sediments, while the further from the shoreline the finer the sediments. When computing the interpolation weights, the algorithm incorporates this natural anisotropy. When interpolating at a point, an observation 100 meters away but in a direction parallel to the shoreline is more likely to be similar to the value at the interpolation point than is an equidistant observation in a direction perpendicular to the shoreline.
Items two, three, and four all incorporate something about the underlying process from which the observations were taken. The length scale, data repeatability, and anisotropy are not a function of the data locations. These enter into the kriging algorithm via the variogram. The length scale is given by the variogram range (or slope), the data repeatability is specified by the nugget effect, and the anisotropy is given by the anisotropy.
Variogram Overview
Surfer includes an extensive variogram modeling subsystem. This capability was added to Surfer as an integrated data analysis tool. The primary purpose of the variogram modeling subsystem is to assist you in selecting an appropriate variogram model when gridding with the kriging algorithm. Variogram modeling may also be used to quantitatively assess the spatial continuity of data even when the kriging algorithm is not applied.
Surfer's variogram modeling feature is intended for experienced variogram users who need to learn Surfer's variogram modeling features. The novice variogram user may find the following four authors helpful: Cressie (1991), Isaaks and Srivastava (1989), Kitanidis (1997), and Pannatier (1996). Please refer to Suggested Reading for full references for each of the previous books. If you do not understand variograms or if you are unsure about which model to apply, use Surfer's default linear variogram with the kriging algorithm.

Variogram modeling is not an easy or straightforward task. The development of an appropriate variogram model for a data set requires the understanding and application of advanced statistical concepts and tools: this is the science of variogram modeling. In addition, the development of an appropriate variogram model for a data set requires knowledge of the tricks, traps, pitfalls, and approximations inherent in fitting a theoretical model to real world data: this is the art of variogram modeling. Skill with the science and the art are both necessary for success.
The development of an appropriate variogram model requires numerous correct decisions. These decisions can only be properly addressed with an intimate knowledge of the data at hand, and a competent understanding of the data genesis (i.e. the underlying processes from which the data are drawn). The cardinal rule when modeling variograms is know your data.
The Variogram
The variogram is a measure of how quickly things change on the average. The underlying principle is that, on the average, two observations closer together are more similar than two observations farther apart. Because the underlying processes of the data often have preferred orientations, values may change more quickly in one direction than another. As such, the variogram is a function of direction.
The variogram is a three dimensional function. There are two independent variables (the direction q, the separation distance h) and one dependent variable (the variogram value g(q,h)). When the variogram is specified for kriging we give the sill, range, and nugget, but we also specify the anisotropy information. The variogram grid is the way this information is organized inside the program.
The variogram (XY plot) is a radial slice (like a piece of pie) from the variogram grid, which can be thought of as a "funnel shaped" surface. This is necessary because it is difficult to draw the three-dimensional surface, let alone try to fit a three dimensional function (model) to it. By taking slices, it is possible to draw and work with the directional experimental variogram in a familiar form - an XY plot.
Remember that a particular directional experimental variogram is associated with a direction. The ultimate variogram model must be applicable to all directions. When fitting the model, the user starts with numerous slices, but must ultimately mentally integrate the slices into a final 3D model.




Minimum Curvature
Minimum Curvature is widely used in the earth sciences. The interpolated surface generated by Minimum Curvature is analogous to a thin, linearly elastic plate passing through each of the data values with a minimum amount of bending. Minimum Curvature generates the smoothest possible surface while attempting to honor your data as closely as possible. Minimum Curvature is not an exact interpolator, however. This means that your data are not always honored exactly.
Minimum Curvature produces a grid by repeatedly applying an equation over the grid in an attempt to smooth the grid. Each pass over the grid is counted as one iteration. The grid node values are recalculated until successive changes in the values are less than the Maximum Residuals value, or the maximum number of iterations is reached (Maximum Iteration field).