CA2002486A1 - Estimation of error of data values with increasing distance from control data points - Google Patents
Estimation of error of data values with increasing distance from control data pointsInfo
- Publication number
- CA2002486A1 CA2002486A1 CA 2002486 CA2002486A CA2002486A1 CA 2002486 A1 CA2002486 A1 CA 2002486A1 CA 2002486 CA2002486 CA 2002486 CA 2002486 A CA2002486 A CA 2002486A CA 2002486 A1 CA2002486 A1 CA 2002486A1
- Authority
- CA
- Canada
- Prior art keywords
- grid
- control data
- data points
- distance
- error
- Prior art date
- Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
- Abandoned
Links
Classifications
-
- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
- G06T17/00—Three dimensional [3D] modelling, e.g. data description of 3D objects
- G06T17/05—Geographic models
Landscapes
- Physics & Mathematics (AREA)
- Engineering & Computer Science (AREA)
- Geometry (AREA)
- Software Systems (AREA)
- Remote Sensing (AREA)
- Computer Graphics (AREA)
- General Physics & Mathematics (AREA)
- Theoretical Computer Science (AREA)
- Image Generation (AREA)
Abstract
ABSTRACT OF THE DISCLOSURE
A method is disclosed 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 have contours rep-resentative of the distance from the control data points.
A residual grid is generated representative of the differ-ence between the gridded surface and a reference surface, which was formed by a separate procedure from the plural-ity 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 dis-tance 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 mimicing the surface.
A method is disclosed 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 have contours rep-resentative of the distance from the control data points.
A residual grid is generated representative of the differ-ence between the gridded surface and a reference surface, which was formed by a separate procedure from the plural-ity 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 dis-tance 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 mimicing the surface.
Description
` 2002486 PATENT
~oller "ESTIMATION OF ERROR OF DATA VALUES WITH
INCREASING DISTANCE FROM CONTROL DATA POINTS"
1. Field of the Invention -The present invention relates to methods of estimating the error of data values with increasing dis-tance from control data points and, more particularly, to 15 such methods which can be utilized with any grid node interpolation algorithm.
. Setting of the Invention Mathematical surfaces can be formed from a plu-rality of control data points with each such data point 20 having a particular value representative of a character-istic. Such characteristics can be elevation above sea level, magnetic susceptibility, magnetic responses, poros-ity, permeability, mineralogy, lithological character-istics, etc. An example of such surfaces are the usual 25 contour elevation maps where elevation lines are drawn between control data points to represent a three-dimen-sional 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 b,een developed which takes control data points and their 5 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 10 algorithms and methodologies employed to generate contour maps in analog or digital form, and a methodology to pro-vide an indication of error.
A grid node interpolation algorithm that is widely used is called Kriging, which provides an estimate 15 of error with distance from the control points. Because the Kriging method provides an estimate of error with dis-tance 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 accu-20 rately represent the surface. Those skilled in the artknow that other grid node/interpolation algorithms, such as moving least squares, moving weighted average and pro-jected slope and the like, could also be used and one or more, in fact, may be better than Kriging. However, these 25 other algorithms have not provided the operator with an indication of the error with increasing distance from con-trol points to provide the operator with a guide to select which algorithm.
~ .. ~ .. .. . .
. . ~ . ,;. -, . . .
- . . ~ .- .:: .. - -- - -.. . . .
:, , :,:, : - - . -: : .
: . .
2002~86 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 5 also a need for a methodolog~ that can evaluate and com-pare the effectiveness of grid node interpolation algo-rithms to allow a user to select the mapping algorithm that will most faithfully reproduce the desired surface.
SUMMARY OF THE INVENTION
The present invention has been designed to over-come the foregoing deficiencies and is complicated 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.
15 In the method, a gridded surface is generated from a plu-rality of control data points utilizing a first grid node interpolation algorithm, again such as Kriging, moving least squa~es, moving weighted average, projected slope, and the like. A distance grid is generated from the plu-20 rality 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 25 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 ,.. ~ ,.. ~ , ........... . . ............. ,.- .
,:, .. : ~................ . : , .
~002486 values representative of the error with increasing dis-tance from control data points for the chosen grid node interpolation algorithm. The results are plotted on an ~c-y axis showing the estimates of error versus distance so 5 l:hat 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 variGus grid node interpolation algo-rithms.
BRIEF DESCRIPTION OF THE DRAWINGS
Figure 1 is a flow diagram of a method of esti-mating 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 sur-face 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 20 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 dif-ference in values between the grids of Figure 2 from 25 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 o~ the grids of Figures 4 and 5, in accord-ance with one method of the present invention.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS
The present invention provides a method of esti-5 mating the error of data values with increasing distancefrom control data points. The method provides a practical alternative to the function in the Kriging grid node interpolation algorithm that provides information concern-ing estimates of error with increasing distance from con-10 trol usually associated with mappable data. Also, themethod provides a means for evaluation and comparison of the effectiveness of various grid node interpolation algo-rithms, such as Kriging, moving squares, moving weighted average, projected slope, etc., to faithfully generate a 15 surface.
As shown in Figure 1, the user selects a refer-ence 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 20 as simple as an N-th order trend surface or it can be a hand-drawn contour map. The reference surface is pre-sented in graphical form and numerous algorithms are available to transform a contour map into a digital grid-ded surface. The reference surface will be referred to 25 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 5 ,above elevation, permeability, porosity, magnetic suscep-tibility, magnetic response, lithology, mineralogy, etc.
Using the control data points, another gridded surface is formed which is referred to as "Grid 2" utiliz-ing the same control data points and a separate grid node 10 interpolation algorithm. Figure 3 is an example of a con-toured moving least squares srid formed from the same con- -trol data points as in Figure 2.
A "Distance Grid" called "Grid 3" is generated from the original control data points with the contours of 15 this grid representative of the distance from the control points in arbitrary units, such for example, 100 ft ele-vation differentials. Figure 4 is an example of a Dis-tance 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 dif-25 ferential.
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 increas-ing distance from control data points. These residuals .~ ~ . , - - .. , , -- :- ~ , - . - - . - .
~ . .
``',.~ , ''' . . : ; ' ' "
200;~86 c:an then be plotted, as shown in Figure 6, to visually represent the mean or absolute residuals against distance f.rom control points again in arbitrary values. Such rep-resentation can be as a hardcopy, such an an x-y plot, in 5 digital form or on a CRT. From this plot the ùser 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 interpo-10 lation algorithm is best to map a surface, the followingprocedure 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. There-15 after, using the same reference grid (Grid 1) in the sameDistance Grid (Grid 33, 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 20 the type shown in Figure 7, that compares how well various grid node interpolation algorithms mimic a reference sur-face with increasing distance from control.
Wherein the present invention has been described in particular relation to the drawings attached hereto, it 25 should be understood that other and further modifications, apart from those shown or suggested herein, may be made within the scope and spirit of the present invention.
.
~oller "ESTIMATION OF ERROR OF DATA VALUES WITH
INCREASING DISTANCE FROM CONTROL DATA POINTS"
1. Field of the Invention -The present invention relates to methods of estimating the error of data values with increasing dis-tance from control data points and, more particularly, to 15 such methods which can be utilized with any grid node interpolation algorithm.
. Setting of the Invention Mathematical surfaces can be formed from a plu-rality of control data points with each such data point 20 having a particular value representative of a character-istic. Such characteristics can be elevation above sea level, magnetic susceptibility, magnetic responses, poros-ity, permeability, mineralogy, lithological character-istics, etc. An example of such surfaces are the usual 25 contour elevation maps where elevation lines are drawn between control data points to represent a three-dimen-sional 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 b,een developed which takes control data points and their 5 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 10 algorithms and methodologies employed to generate contour maps in analog or digital form, and a methodology to pro-vide an indication of error.
A grid node interpolation algorithm that is widely used is called Kriging, which provides an estimate 15 of error with distance from the control points. Because the Kriging method provides an estimate of error with dis-tance 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 accu-20 rately represent the surface. Those skilled in the artknow that other grid node/interpolation algorithms, such as moving least squares, moving weighted average and pro-jected slope and the like, could also be used and one or more, in fact, may be better than Kriging. However, these 25 other algorithms have not provided the operator with an indication of the error with increasing distance from con-trol points to provide the operator with a guide to select which algorithm.
~ .. ~ .. .. . .
. . ~ . ,;. -, . . .
- . . ~ .- .:: .. - -- - -.. . . .
:, , :,:, : - - . -: : .
: . .
2002~86 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 5 also a need for a methodolog~ that can evaluate and com-pare the effectiveness of grid node interpolation algo-rithms to allow a user to select the mapping algorithm that will most faithfully reproduce the desired surface.
SUMMARY OF THE INVENTION
The present invention has been designed to over-come the foregoing deficiencies and is complicated 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.
15 In the method, a gridded surface is generated from a plu-rality of control data points utilizing a first grid node interpolation algorithm, again such as Kriging, moving least squa~es, moving weighted average, projected slope, and the like. A distance grid is generated from the plu-20 rality 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 25 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 ,.. ~ ,.. ~ , ........... . . ............. ,.- .
,:, .. : ~................ . : , .
~002486 values representative of the error with increasing dis-tance from control data points for the chosen grid node interpolation algorithm. The results are plotted on an ~c-y axis showing the estimates of error versus distance so 5 l:hat 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 variGus grid node interpolation algo-rithms.
BRIEF DESCRIPTION OF THE DRAWINGS
Figure 1 is a flow diagram of a method of esti-mating 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 sur-face 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 20 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 dif-ference in values between the grids of Figure 2 from 25 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 o~ the grids of Figures 4 and 5, in accord-ance with one method of the present invention.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS
The present invention provides a method of esti-5 mating the error of data values with increasing distancefrom control data points. The method provides a practical alternative to the function in the Kriging grid node interpolation algorithm that provides information concern-ing estimates of error with increasing distance from con-10 trol usually associated with mappable data. Also, themethod provides a means for evaluation and comparison of the effectiveness of various grid node interpolation algo-rithms, such as Kriging, moving squares, moving weighted average, projected slope, etc., to faithfully generate a 15 surface.
As shown in Figure 1, the user selects a refer-ence 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 20 as simple as an N-th order trend surface or it can be a hand-drawn contour map. The reference surface is pre-sented in graphical form and numerous algorithms are available to transform a contour map into a digital grid-ded surface. The reference surface will be referred to 25 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 5 ,above elevation, permeability, porosity, magnetic suscep-tibility, magnetic response, lithology, mineralogy, etc.
Using the control data points, another gridded surface is formed which is referred to as "Grid 2" utiliz-ing the same control data points and a separate grid node 10 interpolation algorithm. Figure 3 is an example of a con-toured moving least squares srid formed from the same con- -trol data points as in Figure 2.
A "Distance Grid" called "Grid 3" is generated from the original control data points with the contours of 15 this grid representative of the distance from the control points in arbitrary units, such for example, 100 ft ele-vation differentials. Figure 4 is an example of a Dis-tance 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 dif-25 ferential.
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 increas-ing distance from control data points. These residuals .~ ~ . , - - .. , , -- :- ~ , - . - - . - .
~ . .
``',.~ , ''' . . : ; ' ' "
200;~86 c:an then be plotted, as shown in Figure 6, to visually represent the mean or absolute residuals against distance f.rom control points again in arbitrary values. Such rep-resentation can be as a hardcopy, such an an x-y plot, in 5 digital form or on a CRT. From this plot the ùser 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 interpo-10 lation algorithm is best to map a surface, the followingprocedure 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. There-15 after, using the same reference grid (Grid 1) in the sameDistance Grid (Grid 33, 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 20 the type shown in Figure 7, that compares how well various grid node interpolation algorithms mimic a reference sur-face with increasing distance from control.
Wherein the present invention has been described in particular relation to the drawings attached hereto, it 25 should be understood that other and further modifications, apart from those shown or suggested herein, may be made within the scope and spirit of the present invention.
.
Claims (5)
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.
(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) com-prises 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 resi-dual 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) gener-ating a graphical display of the mean residual values for each grid node interpolation algorithm versus distance from control data points.
Applications Claiming Priority (2)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| US38143989A | 1989-07-18 | 1989-07-18 | |
| US381,439 | 1989-07-18 |
Publications (1)
| Publication Number | Publication Date |
|---|---|
| CA2002486A1 true CA2002486A1 (en) | 1991-01-18 |
Family
ID=23505029
Family Applications (1)
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| CA 2002486 Abandoned CA2002486A1 (en) | 1989-07-18 | 1989-11-08 | Estimation of error of data values with increasing distance from control data points |
Country Status (2)
| Country | Link |
|---|---|
| CA (1) | CA2002486A1 (en) |
| GB (1) | GB2234376A (en) |
Families Citing this family (1)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN115794447B (en) * | 2023-02-07 | 2023-04-28 | 青岛哈尔滨工程大学创新发展中心 | Grid data transmission method for multi-physical field coupling |
-
1989
- 1989-11-08 CA CA 2002486 patent/CA2002486A1/en not_active Abandoned
-
1990
- 1990-06-28 GB GB9014427A patent/GB2234376A/en not_active Withdrawn
Also Published As
| Publication number | Publication date |
|---|---|
| GB9014427D0 (en) | 1990-08-22 |
| GB2234376A (en) | 1991-01-30 |
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Legal Events
| Date | Code | Title | Description |
|---|---|---|---|
| EEER | Examination request | ||
| FZDE | Dead |