GB2234376A - Estimation of error of data values with increasing distance from control data points - Google Patents

Abstract

In a method for estimating the error of data values with increasing distance from control data points, a gridded surface is generated from a plurality of control data points, and a distance grid is generated, from the same control data points, that has contours representative of the distance from the control data points. A residual grid is generated representative of the difference between the gridded surface and a reference surface, which was formed by a separate procedure from the plurality of control data points. The distance grid is then integrated with the residual grid to provide mean residual values representative of the error with increasing distance from control data points, and the output is plotted. This method can be utilized with a plurality of grid node interpolation algorithms and the outputs can be plotted side-by-side for the user to determine which grid node interpolation algorithm is best for mimicking the surface. <IMAGE>

Description

"ESTIMATION OF ERROR OF DATA VALUES WITH INCREASING DISTANCE FROM CONTROL DATA POINTS" The present invention relates to methods of estimating the error of data values with increasing distance from control data points and, more particularly, to such methods which can be utilized with any grid node interpolation algorithm.

Mathematical surfaces can be formed from a plurality of control data points with each such data point having a particular value representative of a characteristic. Such characteristics can be elevation above sea level, magnetic susceptibility, magnetic responses, porosity, permeability, mineralogy, lithological characteristics, etc. An example of such surfaces are the usual contour elevation maps where elevation lines are drawn between control data points to represent a three-dimensional surface on a two-dimensional output, such as a hardcopy map, or on a display screen. The process of drawing the contours can be very time consuming if done by hand and is always subject to an error caused by the draw er's estimation of where the lines should be placed.

Numerous computer driven algorithms to map a surface have been developed which takes control data points and their values and create contour maps and/or digital maps formed from a plurality of grid nodes, each having a value.

"Statistics and Data Analysis in Geology," Second Edition, Davis, Kansas Geological Survey, 1985, John Wiley & Sons Publishers describes different types of algorithms and methodologies employed to generate contour maps in analog or digital form, and a methodology to provide an indication of error.

A grid node interpolation algorithm that is widely used is called Kriging, which provides an estimate of error with distance from the control points. Because the Kriging method provides an estimate of error with distance from control points, users may choose Kriging more often than would be beneficial because Kriging may not be the best grid node interpolation algorithm to most accurately represent the surface. Those skilled in the art know that other grid node/interpolation algorithms, such as moving least squares, moving weighted average and projected slope and the like, could also be used and one or more, in fact, may be better than Kriging. However, these other algorithms have not provided the operator with an indication of the error with increasing distance from control points to provide the operator with a guide to select which algorithm.

There is a need for a method of providing a practical alternative to the function in Kriging that pro vides information concerning the estimates of error with increasing distance in the control data points. There is also a need for a methodology that can evaluate and compare the effectiveness of grid node interpolation algorithms to allow a user to select the mapping algorithm that will most faithfully reproduce the desired surface.

The present invention has been designed to overcome the foregoing deficiencies and is intended to meet the above-described needs. Specifically, the present invention is a method of estimating the error of data values with increasing distance from control data points.

In the method, a gridded surface is generated from a plurality of control data points utilizing a first grid node interpolation algorithm, again such as Kriging, moving least squares, moving weighted average, projected slope, and the like. A distance grid is generated from the plurality of the control data points with the contour lines (generated from grid nodes) representing the distance from the control data points. A residual grid is generated representative of the difference between the first gridded surface and a previously generated reference surface also formed from the plurality of control data points. The reference surface can be a hand-drawn map that has been digitized or any form of map or mathematical surface that has been digitized.Thereafter, the distance grid is integrated with the residual grid to provide mean residual values representative of the error with increasing distance from control data points for the chosen grid node interpolation algorithm. The results are plotted on an x-y axis showing the estimates or error versus distance so that the operator can then choose with confidence the estimate of error with increasing distance of control, as well as being able to side by side compare and evaluate the effectiveness of various grid node interpolation algorithms.

The invention will now be described by way of example only, with reference to the accompanying drawings, in which: Figure 1 is a flow diagram of a method of estimating the error of data values with increasing distance from control data points in accordance with the present invention.

Figure 2 is a contoured, third order trend surface being a reference surface used in one method of the present invention.

Figure 3 is an example of a contoured, moving least squares gridded surface used as a second gridded surface in one method of the present invention.

Figure 4 is a distance grid for use in one method of the present invention.

Figure 5 is a residual grid formed from the difference in values between the grids of Figure 2 from Figure 3, in accordance with one method of the present invention.

Figure 6 is a graphical display of mean residual values versus distance from control data points for interpolation of the grids of Figures 4 and 5, in accordance with one method of the present invention.

Figure 7 is a graphical display similar to Figure 6 contrasting the effect of the moving least squares algorithm (MLS) employed in the creation of Figure 3 with the effect of projected slope (PS) moving weighted average (MWA) and Kriging (Krig) algorithms.

The present invention provides a method of estimating the error of data values with increasing distance from control data points. The method provides a practical alternative to the function in the Kriging grid node interpolation algorithm that provides information concerning estimates of error with increasing distance from control usually associated with mappable data Also, the method provides a means for evaluation and comparison of the effectiveness of various grid node interpolation algorithms, such as Kriging, moving squares, moving weighted average, projected slope, etc., to faithfully generate a surface.

As shown in Figure 1, the user selects a reference surface that has been formed from a plurality of given data points, referred to as control data points that have characteristic values. This reference surface can be as simple as an N-th order trend surface or it can be a hand-drawn contour map. The reference surface is presented in graphical form and numerous algorithms are available to transform a contour map into a digital gridded surface. The reference surface will be referred to hereafter as either the "Reference Surface" or the "Grid 1". Figure 2 shows a third ordered trend surface with a plurality of control data points (marked by x's). In the example shown in Figure 2, the control data values show depth below the surface of the earth. However, it should be understood that the data values can be any value of a characteristic that forms a surface.In the particular use of the present invention, the data values have some geological or geophysical significance, such as height above elevation, permeability, porosity, magnetic susceptibility, magnetic response, lithology, mineralogy, etc.

Using the control data points, another gridded surface is formed which is referred to as "Grid 2" utilizing the same control data points and a separate grid node interpolation algorithm. Figure 3 is an example of a contoured moving least squares grid formed from the same control data points as in Figure 2.

A "Distance Grid" called "Grid 3" is generated from the original control data points with the contours of this grid representative of the distance from the control points in arbitrary units, such for example, 100 ft elevation differentials. Figure 4 is an example of a Distance Grid again using the control data points of Figure 2.

A "Residual Grid" called "Grid 4" is formed by subtracting the grid node values of Grid 1 from Grid 2 to generate a grid such as shown in Figure 5 which is a moving least squares residual map with units arbitrarily contoured by the user, such as 100 units of elevation differential.

Thereafter, the Distance Grid (Grid 3) and the Residual Grid (Grid 4) are integrated to provide the mean residual values representative of the error with increasing distance from control data points. These residuals can then be plotted, as shown in Figure 6, to visually represent the mean or absolute residuals against distance from control points again in arbitrary values. Such representation can be as a hardcopy, such an an x-y plot, in digital form or on a CRT. From this plot the user can see how well a particular grid node interpolation algorithm can mimic a given surface with increasing distance from control.

In order to determine which grid node interpolation algorithm is best to map a surface, the following procedure can be used. The steps of the present invention above described can be repeated for the original control data points using any number of grid node interpolation algorithms to generate one grid per algorithm. Thereafter, using the same reference grid (Grid 1) in the same Distance Grid (Grid 3), the latter steps of generating a residual grid and integrating the distance grid with the residual grid for each algorithm to provide mean residual values for each new grid. The resultant will be a plot as the type shown in Figure 7, that compares how well various grid node interpolation algorithms mimic a reference surface with increasing distance from control.

Wherein the present invention has been described in particular relation to the drawings attached hereto, it should be understood that other and further modifications, apart from those shown or suggested herein, may be made within the scope of the present invention as set forth in the accompanying claims.

Claims (6)

1. A method of estimating the error of data values with increasing distance from control data points, comprising: (a) generating a gridded surface from a plurality of control data points; (b) generating a distance grid from the plurality of control data points with contours repre sentative of distance from the control data points; (c) generating a residual grid represen tative of the difference between the gridded surface of step (a) and a reference surface formed from the plurality of control data points; and (d) integrating the distance grid with the residual grid to provide mean residual values repre sentative of the error with increasing distance from control data points.

2. The method of Claim 1 wherein the reference gridded surface is a contour elevational map.

3. The method of Claim 1 wherein step (a) comprises utilizing a grid node interpolation algorithm to generate the distance grid.

4. The method of Claim 1 and including step (e) generating a graphical display of the mean residual values versus distance from control data points.

5. The method of Claim 1 and including step (e) repeating steps (a)-(d) for a plurality of grid node interpolation algorithms of steps (a); and (f) generating a graphical display of the mean residual values for each grid node interpolation algorithm versus distance from control data points.

6. A method of estimating the error of data values with increasing distance from control data points, substantially as hereinbefore described with reference to the accompanying drawings.