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May 9, 2018 | Author: Robby Zafnia | Category: Interpolation, Contour Line, Parameter (Computer Programming), Equations, Regression Analysis
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Surfer Gridding (Golden Software)

Gridding Overview Gridding produces a regularly spaced array of Z values from irregularly spaced XYZ data. Contour maps and Surface plots require the regular regular distribution of data points in grid [.GRD] files. The term "irregularly spaced" implies that the points are randomly distributed over the extent of the map area meaning that the distance between data data points is not consistent over the map. When the XYZ data is randomly spaced over the map area, there are many "holes" in the distribution of data points. Gridding fills in the holes by extrapolating extrapolating or interpolating Z values in those those locations where no data exists. Surfer provides you with several several gridding methods. Each method calculates grid node values using a different algorithm, and can result in a somewhat different interpretation interpretation of your data. Gridding Methods Surfer is a grid based contour program. Gridding is the process of using original data points (observations) in an XYZ data file to generate calculated data points on a regularly spaced grid (a grid [.GRD] file). Interpolation Interpolation schemes estimate the value of the surface at locations where no original data exists, based on the known data values (observations). (observations). Surfer then uses the grid to generate the contour map or surface plot. The advantages of a grid based approach outweigh the disadvantages. disadvantages. Tasks such as drawing contour lines, volumetric calculations of map modifications are much faster with a grid based approach. Under most circumstances, circumstances, there are few problems problems when using a grid file to produce a contour map versus using the original raw data to produce the contour map. One potential disadvantage to gridding is possibility that your original data points might not be honored in the grid file. Contour maps are drawn from the the interpolated grid rather than than the original input data points. There is no guarantee that your original data is honored in the Most of the gridding gridding methods in Surfer use a weighted average interpolation algorithm. algorithm. This means that, with all other factors being equal, the closer a data point is to a grid node, the more weight it carries in determining the Z value at a particular grid node. For more information on selecting a method see Choosing a Gridding Method. Sometimes the Z grid limits are beyond the limits of the original data points. This can happen in areas on the map that are not supported by data such as along the edge of a map or in large holes that contain no original original data. This effect is also a function function of the gridding method method used. Inverse distance cannot generate values outside outside the limits of the original data. Other methods try to define trends in the data. data. When trends are not negated negated by the original data, such such as in areas along the edge of the map, the trends might result in Z values beyond the limits of the original data. If this occurs, you can try a different gridding method, or use the Grid Math command to truncate the grid values at some user-defined limit. Exact and Smoothing Interpolators  The gridding methods included with Surfer can be divided into two general categories: Exact Interpolators and Smoothing Interpolators. Interpolators. Actually some of the methods can fall under either category depending on the options specified for the individual method. Because Surfer contour maps are created from gridded data and not the original raw data, the original data points might not be honored exactly by the grid grid file. For example, if you post the the original data points over the top of a contour map. some of the original data points might be plotted on the "wrong" "w rong" side of a contour. This happens because the averaging of data values might increase or decrease grid node values

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Surfer Gridding (Golden Software)

in the location of an original data point. The grid points are honored exactly, but the input data points might not be honored exactly. Exact interpolators can honor data points exactly only when the data point falls directly on a grid node being interpolated. With weighted average interpolators this means that the coincident data point carries a weight of essentially 1.0 and all other data points carry a weight of essentially zero. Data points are applied directly only when the data point and the grid node are exactly coincident. Even when using exact interpolators it is possible that the data is not honored exactly by the grid file. To reduce the possibility of not honoring original data points, you can increase the number of grid lines in the X and Y direction. This increases the likelihood that the grid nodes directly overlie your data points, thereby increasing the likelihood that the data points are appli ed directly to the grid file. Smoothing interpolators or smoothing factors can be employed during gridding when you do not have strict confidence in your data measurements. Smoothing interpolators do not assign weights of 1.0 to any single data point, even when the data is exactly coincident with the grid node. This doesn't mean that the contour maps are not accurate representations of the data, only that Smoothing interpolators modify the weighting factors in such a way so the surface is smoother; in other words, the weighting factors are spread out more evenly among the data points. In the most extreme case, all data points are given equal weight and the surface assumes the level corresponding to the average for all data in the data file. Recommendations for Choosing a Gridding Method Surfer provides you with a large list of gridding methods and options. Different gridding methods can have different results when interpreting your data. The guidelines presented here should be used as a first approach to deciding which gridding method is best to use with your data. These are only general recommendations, and ultimately you should use the gridding method that produces the map that best represents your data. With most data sets the default gridding method, Kriging with a linear variogram is quite effective. In general this is the method that we would most often recommend. A very close second is Radial Basis Multiquadrics. Either of these methods is likely to produce a reasonable representation of your data. The following list gives you a quick overview of each gridding method and some advantages and disadvantages to selecting one method over another. •



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Inverse Distance is fast but has the tendency to generate "bull's-eye" patterns of concentric contours around the data points. Kriging is one of the more flexible methods and is useful for gridding almost any type of data set. With most data sets, Kriging with a linear variogram is quite effective. In general this is the method that we would most often recommend. Kriging is the default gridding method because it generates the best overall interpretation of most data sets. For larger data sets, however, Kriging can be rather slow. Minimum Curvature generates smooth surfaces and is fast for most data sets. Nearest Neighbor is useful for converting regularly spaced XYZ data files to Surfer grid files. Or when your data is a nearly complete grid with only some missing holes, this method is useful for filling in the holes, or creating a grid file with the blanking value assigned to those locations where no data is present.

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Polynomial Regression processes the data so that underlying large scale trends and patterns are shown. This is used for trend surface analysis. Polynomial Regression is very fast for ny amount of data, but local details in the data are lost in the generated grid. * Radial Basis Functions is quite flexible, and like Kriging, generates among the best overall interpretations of most data sets. This method produces results that are quite similar to Kriging. Shepard's Method is similar to Inverse Distance but does not tend to generate "bull's eye" patterns, especially when a Smoothing factor is used. Triangulation with Linear Interpolation is fast with all data sets. When you use small data sets Triangulation generates distinct triangular facets between data points. One advantage of triangulation is that, with enough data, triangulation can preserve break lines defined in a data file. For example, if a fault is delimited by enough data points on both sides of the fault line, the grid generated by triangulation will show the discontinuity. Inverse Distance to a Power

The Inverse Distance to a Power gridding method is a weighted average interpolator, and can be either an exact or a smoothing interpolator. The Power parameter controls how the weighting factors drop off as distance from a grid node increases. For a larger power, closer data points are given a higher fraction of the overall weight; for a smaller power, the weights are more evenly distributed among the data points. The weight given to a particular data point when calculating a grid node is proportional to the inverse of the distance to the specified power of the observation from the grid node. When calculating a grid node, the assigned weights are fractions, and the sum of all the weights is equal to 1.0. When an observation is coincident with a grid node, the observation is given a weight of essentially 1.0, and all other observations are given a weight of almost 0.0. In other words, the grid node is assigned the value of the coincident observation. This is an exact interpolator. One of the characteristics of inverse distance 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 gridding. A smoothing parameter greater than zero assures that no one observation is given all the weight at a particular grid node, even if the observation is coincident with the grid node. The smoothing parameter reduces the "bull's-eye" effect by smoothing the interpolated grid. Inverse distance is a very fast method for gridding. With less than 500 data points, you can use the All Data search type and gridding will proceed rapidly. Inverse Distance Options  •





The Parameters group box allows you to specify the Power and Smoothing factors to apply during the gridding operation. The Pow er parameter determines how quickly weights fall off with distance from the grid node. As the power parameter approaches zero, the generated surface approaches a horizontal planar surface through the average of all observations from the data file. As the power parameter increases, the generated surface is a "nearest neighbor" interpolator and the resultant surface becomes polygonal. The polygons represent the nearest observation to the interpolated grid node. The Smoothing parameter allows you to incorporate an "uncertainty" factor associated with your input data. The larger the smoothing parameter, the less overwhelming influence any particular observation has in predicting a neighboring grid node.

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Surfer Gridding (Golden Software)





The Anisotropy group box introduces different weighting factors along different anisotropy axes. See Anisotropy for more information. The Data Treatment group box controls the data to be included in the gridding operation. See Data Treatment for more information. Kriging

Kriging is a geostatistical gridding method that has proven useful and popular in many fields. This method produces visually appealling contour and surface plots from irregularly spaced data. Kriging attempts to express trends that are 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. It can be custom fit to a data set by specifying the appropriate variogram model. Within Surfer, Kriging can be either an exact interpolator or a smoothing interpolator depending on the user specified parameters. It incorporates anisotropy and underlying trends in an efficient and natural manner. There are three factors that are uniquely incorporated in the Kriging method: Variogram Model, the Drift Type and the Nugget Effect. These factors can all be controlled from the Kriging Options dialog box. Variogram Model  The variogram model mathematically specifies the spatial variability of the data set and the resulting grid. The interpolation weights, which are applied to data points during the grid node calculations, are direct functions of the variogram model. Surfer allows for a general nested variogram model incorporating three components. Because of this there are more than five hundred possible combinations of variogram models. Each of the three components can be selected from seven common variogram functions: Spherical, Exponential, Linear, Gaussian, Hole-Effect, Quadratic, and Rational Quadratic. Each of the components allow for independent specification of the anisotropy. Computing an experimental variogram from your data is the only certain way to determine which variogram model you should use. A detailed variogram analysis can offer insights into the data that would not otherwise be available, and it allows for an objective assessment of the variogram scale and anisotropy. There are lengthy chapters in many geostatistics textbooks discussing the tools and techniques necessary to generate a variogram (e.g. Isaaks and Srivastava, 1989). When in doubt, you should use the Linear variogram model with the default Scale (C) and Length (A) parameters. With the exception of the Linear variogram model (which does not have a sill), the Scale parameters (denoted by C in the variogram equations) define the sill for the variogram components you select. Thus, the sill of the variogram model equals the Nugget Effect plus the sum of the components Scale (C) parameters. In most situations, the variogram model sill is approximately equal to the variance of the observed data. The Length (A) parameters define how rapidly the variogram components change with increasing separation distance. The Length (A) parameter for a variogram component is used to scale the physical separation distance. For the Spherical and Quadratic variogram functions, the Length (A) parameter is also known as the variogram range.

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With a Linear variogram model, the slope is given by the Scale/Radius. By allowing an anisotropic radius, it is possible to specify an anisotropic linear variogram slope. Drift Type  When the data points are evenly dispersed within the area of interest, the Drift Type option has little effect on the generated grid. The Drift Type option will have a significant effect during gridding when interpolating across large holes in the data distribution pattern, and when extrapolating beyond the limits of the data. Three drift options are available in Surfer: No Drift, Linear Drift, and Quadratic Drift. When in doubt, it is best to use the No Drift option, meaning that the interpolation uses "Ordinary Kriging". No Drift is appropriate when your data is evenly dispersed. The Linear Drift and Quadratic Drift options are used to implement "Universal Kriging". The use of linear or quadratic drift should be based upon knowledge of an underlying trend of the data. If the data tends to vary around a linear trend, then the Linear Drift option is most appropriate. If the data tends to vary around a quadratic trend (e.g. a parabolic bowl), then the Quadratic Drift option is most appropriate. Kriging Options  When you select Kriging as the Gridding Method and click on the Options button, the Kriging Options dialog box is displayed.

* In the Variogram Model group you can specify up to three nested variograms, and the Scale (C) and Length (A) parameters to use for each. If you do not know which variogram type to select, Linear works well in most cases. If you want to be more precise with the variogram type you should generate a variogram based on your data and compare the generated variogram with models of the different type The Scale (C) parameter controls the vertical scale for the variogram. The variogram sill is defined as the Scale plus the Nugget Effect. You can refer to the Surfer Users Guide for more information. You can also define anisotropy for each variogram you specify. Click the Anisotropy button and the Variogram Anisotropy dialog box is displayed. Specify the Ratio and Angle values, and the graphic image indicates the anisotopy ellipse to be applied. Click OK to return to the Kriging Options dialog box. * The Drift Type group box allows you to select the type of drift model to apply during the Kriging operation. You can select from three models. The No Drift selection invokes Ordinary Kriging and is appropriate for Kriging of data sets with a uniformly dense distribution. The Linear Drift and Quadratic Drift selections are most effective on data sets where large holes exist between data points, or where you are extrapolating beyond the limits of your input data. * The Nugget Effect group box is used when there are potential errors in the collection of your data, or when the sample sizes for your data are too small to provide statistical significance. The nugget effect is implied from the semivariogram you generate of your data. Specifying a nugget effect causes Kriging to become more of a smoothing interpolator, implying less confidence in individual data points versus the overall trend of the data. As you increase the nugget effect, data points are not honored as closely.

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The Error Variance edit box allows you to specify the variance of the measurement errors. This value is a quantification of the repeatability of the data measurements. The Micro Variance edit box allows you to specify the variance of the small scale structure. Anisotropy Anisotropy during gridding implies a preferred direction, or direction of higher or lower continuity between data points. Anisotropy is applied by specifying an anisotropy ratio which states: "Give more weighting to points located along one axis versus points located along another axis." Alternatively you can say that points that lie farther away along one axis are given equivalent weights to points that lie closer along the other axis. The relative weighting is defined by the anisotropy ratio. Under most circumstances, you do not need to employ anisotropy during gridding because most maps use X and Y coordinates that are plotted on the same scale. In this case it can be stated that 1 X unit equals 1 Y unit. In this case, anisotropy ratios are not appropriate and do not have to be applied. There are cases where it is not desirable to assign the same scale to X and Y units when you are creating a grid [.GRD] file. Anisotropy refers to different properties in different directions. During gridding anisotropy implies that different weighting factors are applied in different directions. Anisotropy is applied by specifying an anisotropy ratio which states: "Give more weighting to points located along one axis versus points located along another axis." Alternatively you can say that points that lie further away along one axis are given equivalent weights as points that lie closer along the other axis. The relative weighting is defined by the anisotropy ratio. Using Anisotropy  For each different gridding method, anisotropy might be specified in a slightly different manner. When you create a grid, the Scattered Data Interpolation dialog box is displayed. To assign anisotropy: 1. In the Gridding Method group, select the gridding method from the drop-down list box. 2. Click the Options button and the Options dialog box for the selected gridding method is displayed. 3. In the Anisotropy group, specify the anisotropy parameters. In the Kriging Options dialog box, click the Anisotropy button to display the Variogram Anisotropy dialog box. For different methods, the anisotropy parameters might be somewhat different. As you define the anisotropy parameters, the graphic image in the group indicates the orientation and relative ratios. * Ratio values are defined for Inverse Distance, Kriging, Minimum Curvature, Radial Basis Functions, and Triangulation. A Ratio of one is the default setting, and makes the gridding isotropic. To apply anisotropy, either increase or decrease the Ratio value. * For Shepard's Method, you specify Range values that are used to apply higher or lower relative weighting along the specified axis. Range values are expressed in data units. To apply higher relative weighting along a particular axis you can increase the Range value for that axis, or you can decrease the Range value for the opposing axis. 4. The Angle edit box defines the orientation of the anisotropy axes. Angle is available for all methods except Minimum Curvature.

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Surfer Gridding (Golden Software)

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 is not always honored exactly. When you select Minimum Curvature as the Gridding Method and click on the Options button, the Minimum Curvature Options dialog box is displayed.

* The Parameters group box allows you to control the convergence criteria for Minimum Curvature. The Max Residuals parameter has the same units as the data, and an appropriate value is approximately 10% of the data precision. If data values are measured to the nearest 1.0 units, the Max Residuals value should be set at 0.1. The Iterations continue until the maximum grid node correction for the entire iteration is less than the Max Residuals value. The Max Iterations parameter should be reasonably set at one to two times the number of grid nodes generated in the grid file. For example, when generating a 50 by 50 grid using Minimum Curvature, the Max Iterations value should be set between 2,500 and 5,000. Nearest Neighbor The Nearest Neighbor gridding method assigns the value of the nearest datum point to each grid node. This method is useful when data is already on a grid, but needs to be converted to a Surfer grid file. Or, in cases where the data is nearly on a grid with only a few missing values, this method is effective for filling in the holes in the data. Sometimes with nearly complete grids of data there are areas of missing data that you want to exclude from the grid file. In this case you can set the Search Ellipse to a small value so the areas of no data are assigned the blanking value in the grid file. By setting the search ellipse radii to values less than the distance between data values in your file, the blanking value will be assigned at all grid nodes where no datum values exist. When you want to use the Nearest Neighbor method to convert regularly spaced XYZ data to a grid file, you can set the grid spacing equal to the spacing between data points in the file. When you select Nearest Neighbor as the Gridding Method and click on the Options button, the Nearest Neighbor Options dialog box is displayed. Polynomial Regression Polynomial Regression is used to define large scale trends and patterns in your data. There are several options you can use to define the type of trend surface you want. Polynomial Regression is not really an interpolator because it does not attempt to predict unknown Z values. * The Surface Definition group box allows you to select the type of polynomial regression you want to apply to your data. As you select the different types of polynomials, a generic polynomial form of the equation is presented in the group box, and the values in the Parameters group box change to reflect your selection. The available choices are: Simple planar surface Bi-linear saddle Quadratic surface

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Surfer Gridding (Golden Software)

Cubic surface The User defined polynomial allows you to define polynomial equations by utilizing the Parameters options. * The Parameters group box allows you to specify the maximum powers for the X and Y component in the polynomial equation. As you change the Parameters values, the options are changed in the Surface Definition group box to reflect the parameters you define. The Max X Order specifies the maximum power for the X component in the polynomial equation. The Max Y Order specifies the maximum power for the Y component in the polynomial equation. The Max Total Order specifies the maximum sum of the Max X Order and Max Y Order  powers. All of the combinations of the X and Y components are included in the polynomial equation as long as the sum of the two powers does not exceed the Max Total Order value. * The Data Treatment group box controls the data to be included in the gridding operation. See Data Treatment for more information. * The Copy the regression coefficients to the clipboard checkbox sends a copy of the coefficients in the Windows Clipboard. When you grid the file, the coefficients are calculated and used in the calculation of the surface, and are written to the clipboard. Radial Basis Functions Radial Basis Functions are a diverse group of data interpolation methods. In terms of the ability to fit your data and to produce a smooth surface, the Multiquadric method is considered by many to be the best method. All of the Radial Basis Function methods are exact interpolators so they make an attempt to honor your data. You can introduce a smoothing factor to all the methods in an attempt to produce a smoother surface. The functions you can specify are analogous to variograms in Kriging. The functions define the optimal set of weights to apply to the data points when interpolating a grid node. Radial Basis Function Options  * The Basis Function group box allows you to specify the function parameters for the gridding operation The Type drop-down list box assigns function to use during gridding. These define the optimal weights applied to the data points during the interpolation. The basis functions are analogous to variograms in Kriging. Under most circumstances, the Multiquadric function is the most desirable. The R2 parameter is a shaping or smoothing parameter. The larger the R2 parameter, the rounder the mountain tops and the smoother the contour lines. There is no universally accepted method for computing an optimal value for this parameter. A reasonable trial value for R2 is between the average sample spacing and one-half the average sample spacing. Triangulation w/ Linear Interpolation The Triangulation interpolator is an exact interpolator. The method works by creating triangles by drawing lines between data points. The original data points are connected in such a way that no triangle edges are intersected by other triangles. The result is a patchwork of triangular faces over the extent of the grid. Each triangle defines a plane over the grid nodes lying within the triangle, with the tilt and elevation of the triangle determined by the three original data points defining the triangle. All grid nodes within a given triangle are defined by the triangular surface. Because the original data points are used to define the triangles, your data is honored very closely.

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Surfer Gridding (Golden Software)

Triangulation works best when your data points are evenly distributed over the grid area. Data sets that contain sparse areas result in distinct triangular facets on a surface plot or contour map. Triangulation is very effective at preserving break lines.

Shepard's Method Shepard's Method uses an inverse distance weighted least squares method. As such it is similar to the Inverse Distance to a Power interpolator but the use of local least squares eliminates or reduces the "bull's eye" appearance of the generated contours. Shepard's Method can be either an exact or a smoothing interpolator. When you select Shepard's Method as the Gridding Method and click on the Options button, the Modified Shepard's Method dialog box is displayed. * The Parameter group box assigns a smoothing parameter to the gridding operation. The Smoothing parameter allows Shepard's Method to operate as a smoothing interpolator. As you increase the value of the smoothing parameter, the greater the effect of smoothing. In general, values between zero and one are most reasonable. * The Anisotropy group box introduces different weighting factors along different anisotropy axes. See Anisotropy for more information. * The Data Treatment group box controls the data to be included in the gridding operation. See Data Treatment for more information. * The Reset button returns the parameters to the default values used for the data set to be gridded. Smoothing a Grid File The smoothing commands operate on grid files. Smoothing operations are used to even out angular contours and blocky surfaces, or to eliminate noise in a contour map or surface. Spline Smoothing is most effective for eliminating angular contours or surfaces by filling in a sparse grid. For example, a 10x10 grid might result in very angular contours on a contour map produced from the grid. By increasing the grid density using the spline smoothing command, the 10x10 grid can result in a 50x50 grid. The 50x50 grid would produce a much smoother appearing contour map than the 10x10 grid. Matrix Smoothing is most effective for removing noise or variability between grid nodes that are close together. The general trends in the surface are retained, but spikes tend to be eliminated. Matrix smoothing results in an output grid with the same dimensions as the input grid. Matrix smoothing generates a blanked border around the outside of the grid.

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