The Minimum Curvature method is a fast, versatile interpolation method that produces an optimally smooth surface that is visually appealing. The method is equally suited to evenly spaced data, line oriented data and randomly spaced data. The implementation largely follows the methodology described by Smith and Wessel (Geophysics 1990).
The interpolated surface is analogous to a thin elastic plate passing through each of the data values. The surface will have continuous second derivatives and minimized total squared curvature. However, like any splining method, it can produce significant overshoots and undershoots which may extend beyond the limits found in the source data. When encountered, these oscillations can be partially controlled by introducing additional tension to the elastic plate equations. The method attempts to honor the input data values, but in practice this is rarely achieved and users should expect some mismatch between the generated surface and the input data values.
The elastic plate equations are solved iteratively. In general, the more iterations the algorithm performs, the closer the solution will be to the optimal solution. However, each iteration is computationally expensive. The algorithm uses techniques to maximize the speed at which the solution converges to the optimal solution and provides controls to users to limit the number of iterations performed.
The method will interpolate a value at all grid cells by default but where there is no input data the equations are unconstrained and clipping ought to be employed to invalidate grid cells. When using minimum curvature, careful consideration ought to be given to the raster cell size. As a good rule of thumb, the cell size should be approximately ¼ of the average data point separation. If the cell size is too small the method may produce ‘pimples’ at each input data point.
Minimum Curvature Method Options
The minimum curvature method offers the following options to control the process and surface.
Parameter Unit
Specifies whether spatial parameters are defined in cell units or distance units. The units should be supported by MapInfo Pro, for example, the options for Distance unit include: US Survey feet, yards, rods, chains, miles, nautical miles, millimeters, centimeters, meters, inches, links and kilometers. If you choose Distance, you need to select a distance unit from the Distance Unit drop-down list.
Iteration Count
The Iteration Count denotes the number of times the equation is applied over the data to produce a smoothly varying raster surface. In each iteration, every cell of the raster is processed to better satisfy the smoothness constraints, and the surface becomes closer to the optimal solution. However, the computation cost of each iteration is large and proportional to the number of cells in the raster.
You can perform as many iterations as you want. You may have to perform many times for a small raster, but fewer for a very large raster. The iteration count option gives you the ability to limit the total number of iterations that may be performed. The iteration count can be set to the following values.
- Minimum
- Very Low
- Low
- Normal
- High
- Very High
- Maximum
Click More Options to open the following advanced options.
Percentage Change
In each iteration, the maximum change in every raster cell is recorded as a percentage of the dynamic range of the source data. As the surface becomes smooth, the maximum change may become very small. This can be used to stop the algorithm from iterating further and to reduce the overall runtime. By setting the percentage change to zero, you can iterate until the iteration count is reached. Set the percentage change to greater than zero, if you want to terminate the algorithm after you have achieved a suitably smooth surface. The larger the percentage change, the faster the algorithm will terminate. The smaller the percentage change, the smoother the final surface.
Spline Tension
Specify a value between 0 and 1 to apply tension to the splines. The default value is is 0. A smaller setting will result in lower overall curvature, or smoother surfaces. However, this can result in undershoots and overshoots in regions where there is no data and the interpolator is unconstrained. A higher setting will result in less smooth surfaces but tends to reduce the amount of undershoot and overshoot. When gridding terrain or bathymetry data it is beneficial to use a moderate level of spline tension (0.25).
Stamping Method
Within each raster cell, only one source data point can be used as input to the elastic plate equations. If two or more source data points lie inside a raster cell then a methodology must be employed to either select which source data value is used or combine them into a single source value.
- First Only - The first data point loaded is assigned to the cell. All other source data points that lie inside the cell are discarded and do not contribute to the solution.
- Last Only - The last data point loaded is assigned to the cell. All other source data points that lie inside the cell are discarded and do not contribute to the solution.
- Average - The value and position of all data points inside the cell are averaged to create a single, fabricated data point.
- Average (Last in Weighted) - The value and position of all data points inside the cell are combined to create a single, fabricated data point. The combination function gives highest weighting to the last data points loaded.
- Average (Inverse Distance Weighted) - The value and position of all data points inside the cell are combined to create a single, fabricated data point. The combination function uses inverse distance based weighting.
Smoothing Method
You can apply smoothing on the gridded data to produce smoother surface. Smoothing is used to enhance the sharpness of an image or improve the appearance of the edges. Select a suitable Smoothing method and level for your data.
Smoothing Level - Move the smoothing slider to set the smoothing level for the output raster. You can set a value between 0 to 6. A value of zero applies no smoothing. A value of 6 applies maximum smoothing.
Clipping
By default the algorithm populates every cell in the raster. You may want to clip the raster to make unconstrained cells invalid. You could do this, for example, by clipping the raster to a boundary polygon. As an alternative, the minimum curvature method provides a fast and simple automated clipping method.
This method allows you to define what an unconstrained cell is by specifying a ‘near’ and ‘far’ distance (as an integer number of cells). Any cell that has a constrained cell inside the ‘near’ distance will be considered constrained. Otherwise, any cell that has constrained cells within the ‘far’ distance on three or more sides will be considered constrained. This method tends to interpolate across voids within the data while terminating on the edges of the data.
- None - No clipping is applied to the raster cells.
- Near Only - The Near value represents the maximum distance from a source input data point for which an interpolated raster cell will be created. Cells in the raster which lie at a distance greater than the Near distance will be assigned a null value. This method has the same effect as applying a distance buffer to the source data points equal to the near distance.
- Near/Far - Interpolated cells in the output raster will be clipped to the near distance if no other data point is found within the Far distance that meet the angular search constraints. Applying both Near and Far clipping can be useful to constrain the interpolated raster to a required distance from the source points, while also permitting larger gaps to be interpolated in irregularly spaced data.
- Polygon - You can provide a TAB file of polygon(s) to clip the output raster to the polygon boundaries. You can specify whether to clip a region outside or inside the raster bounds. However, it does not support polygons with holes.
Cell Size
Specify the cell size for the output raster in the Cell Size box. The cell size defines the width and height of a raster cell in distance units. If the raster cells are square both width and height are specified with same value.
By default, Automatic is selected which means will calculate the output raster cell size based on source data points. Click Suggest, to see the calculated cell size value in the box before it is processed. You can modify the cell size value to produce output raster with the desired cell size. The Suggest button is active only when the input data source is in MapInfo native format.
Output Geometry
The Output Geometry allows you to limit the output data points according to specified region (bounds) and ignore all points that lie outside of the specified region. The data within the specified bounds will be written in the output file. To specify the Raster Bounds, enter coordinate values for raster origin and extent.
- Min X - X coordinate of the origin (lower left corner of the cell).
- Min Y - Y coordinate of the origin (lower left corner of the cell).
- Max X - The maximum coordinate value for X (upper right corner of the cell).
- Max Y - The maximum coordinate value for Y (upper right corner of the cell).
The output file will contain data for the specified region only.
If required, click More Options to specify the projection for the output raster. If the input file is a MapInfo .TAB file, projection values are read from the input file, which you can override here.
- Category - The Category drop-down list consists of all projection systems supported by MapInfo Pro. For example, Longitude/Latitude, Universal Transverse Mercator (ED 50), Universal Transverse Mercator (NAD 27 for Canada), etc.
- Sub Category - The Sub Category drop-down list consists of the type of projection based on the selected projection system.
If required, click More Options to specify category and sub-category of the output raster projection. If the input file is MapInfo .TAB, projection values are read from the input file, which you can override here.
- Category - The Category drop-down list consists of all projection systems supported by MapInfo Pro. For example, Longitude/Latitude, Universal Transverse Mercator (ED 50), Universal Transverse Mercator (NAD 27 for Canada), etc.
- Sub Category - The Sub-Category drop-down list consists of the type of projection based on the selected projection system.