A quasi-geoid precision grid model generation strategy and method

Abstract

This invention provides a strategy and method for generating a geoid-like precision grid model. S1: Using the "cyclic empty point method," the "elevation anomaly residual" of GNSS leveling points is calculated. S2: Each grid point of the gravity geoid model is visited sequentially, and GNSS leveling points within a certain range around it are searched. The distance between the current grid point and each searched GNSS leveling point is calculated. S3: Based on the distances between these GNSS leveling points and the current grid point, and the "elevation anomaly residual" of the GNSS leveling points calculated in the previous step, the precision of the current grid point is calculated, resulting in a precision grid model with the same resolution as the geoid model. This method can provide a calculation method for the precision of specific local areas, overcoming the problem that previous geoid model precision evaluations were not refined enough, only providing an overall model precision and not specific local area precision, thus failing to meet engineering application needs.

Description

Technical Field

[0001] This invention belongs to the field of surveying and mapping methods, specifically relating to a strategy and method for generating a geoid-like precision grid model. Background Technology

[0002] The quasi-geoid is the starting surface for my country's elevation datum and a fundamental condition for the application of space technology and information services. Combining the quasi-geoid with GNSS positioning technology allows for the efficient and economical acquisition of normal elevations with physical meaning and engineering value, modernizing elevation measurement methods. This elevation datum maintenance method is widely used in various industries such as land planning, urban construction, transportation, water conservancy projects, and agricultural development.

[0003] The currently accepted technical process for constructing a quasi-geoid model in the industry is as follows: First, based on dense gravity data and digital elevation models within and around the project area, combined with a global high-order gravity field model, a gravity quasi-geoid is constructed using geophysical methods. Second, control points within and around the project area that have been simultaneously observed using GNSS and leveling, and have measured elevation anomalies, are uniformly diluted into two parts with an 8:2 ratio. 80% of these points are used as fitting points, and 20% are used as check points. Third, the gravity quasi-geoid is fitted and corrected using the fitting points to obtain a preliminary quasi-geoid model. Fourth, the preliminary quasi-geoid is checked for accuracy using the check points, providing an overall model fitting accuracy. Fifth, the gravity quasi-geoid is fitted and corrected using all fitting points and check points to obtain the final quasi-geoid model. The accuracy of this model is the accuracy value calculated in step four.

[0004] While the fitting accuracy index of the quasi-geoid model under the current technical system reliably reflects the overall fitting accuracy of the quasi-geoid model, it does not consider the differences in accuracy within local areas of the model. Influenced by basic data such as topography, gravity, and geoid anomalies, the accuracy of the quasi-geoid model varies across different regions. Therefore, evaluating the accuracy of geoid anomalies in a specific local area of ​​the model based solely on its overall fitting accuracy index is not entirely scientific.

[0005] The method of determining normal height using a quasi-geoid combined with GNSS positioning technology involves the user providing the geodetic coordinates (latitude, longitude, and geodetic height) of a point to be determined. The geoid undulation value of that point is obtained through interpolation using a quasi-geoid model of the area where the point is located. The normal height, which has physical significance, is obtained by subtracting the geometrically significant geodetic height from the geoid undulation value. While obtaining the normal height, users also desire the accuracy of the geoid undulation value in the local area where the point is located to provide an applicability evaluation reference for engineering applications. Unfortunately, current methods for evaluating the accuracy of quasi-geoid models cannot provide the accuracy of specific local models, only the overall model accuracy. This is a technical bottleneck in obtaining normal height using quasi-geoid combined with GNSS positioning technology and urgently needs to be addressed. Summary of the Invention

[0006] To address the problem that existing geoid models only provide an overall accuracy evaluation value, which cannot reflect the actual differences in accuracy between different local areas of the model and thus cannot meet users' actual needs for the accuracy values ​​of specific local areas.

[0007] The purpose of this invention is to provide a strategy and method for generating a geoid-like grid model, comprising the following steps: S1. Using the "cyclic empty point method", the difference between the "measured elevation anomaly value" of each GNSS level point and the "model elevation anomaly value" obtained by interpolation based on the geoid model is calculated, which is the "elevation anomaly residual" of the GNSS level point. S2. Visit each grid point of the gravity-like geoid model one by one, search for GNSS leveling points within a certain range around it, and calculate the distance between the current grid point and each searched GNSS leveling point. S3. Based on the distances between these GNSS leveling points and the current grid point, and the "elevation anomaly residuals" of the GNSS leveling points calculated in the previous step, calculate the accuracy of the current grid point to obtain an accuracy grid model consistent with the resolution of the geoid model.

[0008] Furthermore, S1, employing the "cyclic empty point method," calculates the difference between the "measured elevation anomaly value" and the "model elevation anomaly value" obtained by interpolation based on the quasi-geoid model for each GNSS leveling point. Specifically, the "elevation anomaly residual" of the GNSS leveling point includes the following steps: S21. Among N GNSS leveling points, one point is reserved as a check point each time, and the other N-1 points are used as fitting points. S22. Subtract the normal height from the current check point to obtain its "measured elevation anomaly value"; in, Let be the "measured elevation anomaly" of the i-th GNSS leveling point. Let be the geodetic height of the i-th GNSS leveling point. This represents the normal elevation of the i-th GNSS leveling point; S23. Correct the gravity-based geoid model using the fitted points to obtain a fitted geoid model. The fitting steps are as follows: S231. Using bilinear interpolation, calculate the elevation anomaly values ​​of the fitted points based on the gravity-based geoid model. ; S232. Calculate the elevation outlier correction for each fitted point. ,Right now - ; S233. Visit each grid point in the gravity-based geoid model one by one, and search for at least 8 fitting points around it. Based on the elevation anomaly corrections of these fitting points, use the inverse distance weighting method to calculate the elevation anomaly correction of the grid point. Add this correction to the grid point value to obtain the fitted elevation anomaly of the grid point. After all grid points have been calculated, the fitted geoid model is obtained. Inverse distance weighting method: The coordinates of a certain grid point G in a geoid model are as follows: GNSS leveling points around it Coordinates are Then the elevation anomaly correction for grid point G is: In the formula, The number of GNSS leveling points. Let be the "elevation anomaly correction" for the i-th GNSS leveling point; The weight of the i-th GNSS leveling point is defined as follows: in, For grid point G and GNSS leveling point distance, This is the equal-weighted radius; that is, when the distance is less than this value, this value is used instead of the actual distance. The search radius; S24. Use bilinear interpolation to calculate the elevation anomaly value of the current check point in the fitted geoid grid model, i.e., the "model elevation anomaly value"; S25. Calculate the difference between the "measured elevation anomaly" and the "model elevation anomaly" at the current check point, i.e., the "elevation anomaly residual"; repeat the previous process to calculate the "elevation anomaly residual" for each GNSS leveling point. Let be the "elevation anomaly residual" of the i-th GNSS leveling point. Let be the "model elevation anomaly" of the i-th GNSS leveling point.

[0009] S2 involves sequentially visiting each grid point of the gravity-based geoid model, searching for GNSS leveling points within a certain radius of the current grid point, and calculating the distance from the current grid point to each of the searched GNSS leveling points. The distance calculation formula is as follows: In the formula, B1 and L1 are the latitude and longitude of the grid points, in °; B2 and L2 are the latitude and longitude of the GNSS leveling points, in °; and S is the distance, in km.

[0010] Furthermore, the specific process of S3, which calculates the accuracy of the current grid point based on the distance between these GNSS leveling points and the current grid point and the "elevation anomaly residual" of the GNSS leveling points calculated in the previous step, to obtain an accuracy grid model consistent with the resolution of the geoid model, is as follows: The coordinates of a certain grid point G in a geoid model are as follows: GNSS leveling points around it Coordinates are Then the precision of grid point G is: In the formula, The number of GNSS leveling points. Let be the "elevation anomaly residual" of the i-th GNSS leveling point; The weight of the i-th GNSS leveling point is defined as follows: in, For grid point G and GNSS leveling point distance, This is the equal-weighted radius; that is, when the distance is less than this value, this value is used instead of the actual distance. The search radius is denoted as ; this method is called the inverse distance weighted method.

[0011] This invention provides a strategy and method for generating a geoid-like grid model, which can provide a method for calculating the accuracy of a specific local area model. This overcomes the problem that previous geoid-like models were not precise enough in their accuracy evaluation, and could only provide an overall model accuracy, but could not provide the accuracy of a specific local area model, thus failing to meet the needs of engineering applications.

[0012] The present invention will now be described in detail with reference to the accompanying drawings and embodiments. Attached Figure Description

[0013] Figure 1 This is a diagram illustrating technical terms.

[0014] Figure 2 This is a schematic diagram of the bilinear interpolation algorithm.

[0015] Figure 3 This is a map showing the distribution of GNSS leveling points in the test area.

[0016] Figure 4 It is a geoid-like precision grid contour map of the test area.

[0017] Figure 5 This is a statistical map of the grid points representing the accuracy of the geoid in the experimental area. Detailed Implementation

[0018] To further illustrate the technical means and effects of the present invention in achieving its intended purpose, the specific implementation methods, structural features and effects of the present invention are described in detail below with reference to the accompanying drawings and embodiments.

[0019] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0020] In the description of this invention, it should be understood that the terms "center", "upper", "lower", "front", "rear", "left", "right", "vertical", "horizontal", "aligned", "overlapping", "bottom", "inner", "outer", etc., indicate the orientation or positional relationship based on the orientation or positional relationship shown in the accompanying drawings. They are only for the convenience of describing this invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, they should not be construed as limitations on this invention.

[0021] The terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of technical features indicated. Thus, a feature defined as "first" or "second" may explicitly or implicitly include one or more of that feature; in the description of this invention, unless otherwise stated, "a plurality of" means two or more. Introduction to basic concepts

[0022] like Figure 1 As shown: (1) Reference ellipsoid: The surface of an ellipsoid that is close to the size and shape of the Earth and has been positioned for processing geodetic results.

[0023] (2) Geoid: A closed gravity equipotential surface that coincides with the mean sea level and extends into the interior of the continent. Due to the unevenness of the Earth's surface and the uneven distribution of mass inside the Earth, the geoid is a slightly undulating irregular curved surface. The determination of the geoid involves the density inside the Earth and cannot be directly determined by ground measurement data.

[0024] (3) Quasi-geoid: It coincides with the geoid on the ocean surface and is close to but not completely coincident with the geoid on the continent. It is an auxiliary surface used for calculation and can be directly determined by ground measurement data.

[0025] (4) Quasi-geoid model: A grid model that stores point values. The point value represents the distance between the reference ellipsoid and the quasi-geoid at the current location, that is, the difference between the geoid height at the current location and the normal height.

[0026] (5) Geodetic height: The distance from a ground point to the reference ellipsoid along the normal to the reference ellipsoid, obtained by GNSS observation.

[0027] (6) Normal height: The distance from a ground point along the vertical direction to the geoid, obtained by leveling observation.

[0028] (7) Height anomaly: The height difference between the geoid and the reference ellipsoid. If GNSS and leveling observations are conducted at a point at the same time, the geodetic height and normal height of that point can be obtained. The measured height anomaly of that point is the difference between its geodetic height and normal height.

[0029] (8) GNSS observation: An observation method that obtains three-dimensional coordinates, velocity and time information of any location on the Earth's surface or in near-Earth space through the Global Navigation Satellite System.

[0030] (9) Leveling observation: a method of measuring the height difference between two points on the ground using a level and a leveling rod. Introduction to Grid Model Format

[0031] A grid model refers to a set of point values ​​stored in a fixed-interval order within a rectangular geographic area, stored as text. In this scheme, the gravity quasi-geoid model, the quasi-geoid model, and the quasi-geoid precision grid model are all rectangular grid models with a consistent format. Assuming the rectangular geographic area is latitude 30°~40° and longitude 100°~105°, with an interval of 2′ between adjacent grid points, the specific format is as follows: 40.00 ------ 30.00 2.00 100.00 ------ 105.00 2.00 1.3 2.1 1.7 2.5 3.0 1.9 2.8 1.6 1.9 2.1 ... 2.4 3.1 1.5 2.2 2.0 2.9 1.8 2.3 1.4 2.0 1. The data in the first row is the latitude range of 30º00′-40º00′ and the latitude interval of 2.0′.

[0032] 2. The second row of data is the longitude range of 100º00′-105º00′ and the longitude interval of 2.0′.

[0033] 3. The data from line 3 to the end of the file is grid data, and the grid data within the entire range is arranged in descending order of latitude and ascending order of longitude.

[0034] (1) This data is the data of the grid center point; (2) Grid data is first obtained from the maximum latitude value. Start calculating the latitude of the center point Then the longitude starts from the minimum value Begin by calculating the longitude of the grid center point, arranging the grid columns in ascending order. until the maximum longitude of the grid is reached. The output ends with the longitude of the grid center point. Each row contains 10 columns of data. If there are more than 10 columns, start a new row and arrange the data in ascending order of longitude. If there are fewer than 10 columns of data, output all the data within the longitude range and pad the missing parts with spaces.

[0035] (3) From latitude Minimum latitude of grid Repeat the previous step until all grid data has been arranged. Introduction to Grid Model-Based Interpolation Algorithms

[0036] The interpolation of the target position in the grid model is calculated using a bilinear interpolation algorithm. First, the target position is located in which cell of the grid model it falls into, and the values ​​of the four corner points of that cell are obtained. Then, the bilinear interpolation method is used to calculate the interpolation of the target position based on the values ​​of the four corner points.

[0037] like Figure 2 As shown, the principle of bilinear interpolation is as follows: Assumption point The values ​​of the four nearest corner points are respectively , , , ,but Point-based interpolation based on grid model The calculation formula is:

[0038]

[0039]

[0040] In the formula, , Grid spacing; , For point Distance to grid points The distance.

[0041] Example 1

[0042] To address the problem that existing geoid models only provide an overall accuracy evaluation value, which cannot reflect the actual differences in accuracy between different local areas of the model and thus cannot meet users' actual needs for the accuracy values ​​of specific local areas.

[0043] This embodiment provides a strategy and method for generating a geoid-like grid model with high accuracy, including the following steps: S1. Using the "cyclic empty point method", the difference between the "measured elevation anomaly value" of each GNSS level point and the "model elevation anomaly value" obtained by interpolation based on the geoid model is calculated, which is the "elevation anomaly residual" of the GNSS level point. S2. Visit each grid point of the gravity-like geoid model one by one, search for GNSS leveling points within a certain range around it, and calculate the distance between the current grid point and each searched GNSS leveling point. S3. Based on the distances between these GNSS leveling points and the current grid point, and the "elevation anomaly residuals" of the GNSS leveling points calculated in the previous step, calculate the accuracy of the current grid point to obtain an accuracy grid model consistent with the resolution of the geoid model.

[0044] Furthermore, S1, employing the "cyclic empty point method," calculates the difference between the "measured elevation anomaly value" and the "model elevation anomaly value" obtained by interpolation based on the quasi-geoid model for each GNSS leveling point. Specifically, the "elevation anomaly residual" of the GNSS leveling point includes the following steps: S21. Among N GNSS leveling points, one point is reserved as a check point each time, and the other N-1 points are used as fitting points. S22. Subtract the normal height from the current check point to obtain its "measured elevation anomaly value"; in, Let be the "measured elevation anomaly" of the i-th GNSS leveling point. Let be the geodetic height of the i-th GNSS leveling point. This represents the normal elevation of the i-th GNSS leveling point; S23. Correct the gravity-based geoid model using the fitted points to obtain a fitted geoid model. The fitting steps are as follows: S231. Using bilinear interpolation, calculate the elevation anomaly values ​​of the fitted points based on the gravity-based geoid model. ; S232. Calculate the elevation outlier correction for each fitted point. ,Right now - ; S233. Visit each grid point in the gravity-based geoid model one by one, and search for at least 8 fitting points around it. Based on the elevation anomaly corrections of these fitting points, use the inverse distance weighting method to calculate the elevation anomaly correction of the grid point. Add this correction to the grid point value to obtain the fitted elevation anomaly of the grid point. After all grid points have been calculated, the fitted geoid model is obtained. Inverse distance weighting method: The coordinates of a certain grid point G in a geoid model are as follows: GNSS leveling points around it Coordinates are Then the elevation anomaly correction for grid point G is: In the formula, The number of GNSS leveling points. Let be the "elevation anomaly correction" for the i-th GNSS leveling point; The weight of the i-th GNSS leveling point is defined as follows: in, For grid point G and GNSS leveling point distance, This is the equal-weighted radius; that is, when the distance is less than this value, this value is used instead of the actual distance. The search radius; S24. Use bilinear interpolation to calculate the elevation anomaly value of the current check point in the fitted geoid grid model, i.e., the "model elevation anomaly value"; S25. Calculate the difference between the "measured elevation anomaly" and the "model elevation anomaly" at the current check point, i.e., the "elevation anomaly residual"; repeat the previous process to calculate the "elevation anomaly residual" for each GNSS leveling point. Let be the "elevation anomaly residual" of the i-th GNSS leveling point. Let be the "model elevation anomaly" of the i-th GNSS leveling point.

[0045] S2 involves sequentially visiting each grid point of the gravity-based geoid model, searching for GNSS leveling points within a certain radius of the current grid point, and calculating the distance from the current grid point to each of the searched GNSS leveling points. The distance calculation formula is as follows: In the formula, B1 and L1 are the latitude and longitude of the grid points, in °; B2 and L2 are the latitude and longitude of the GNSS leveling points, in °; and S is the distance, in km.

[0046] Furthermore, the specific process of S3, which calculates the accuracy of the current grid point based on the distance between these GNSS leveling points and the current grid point and the "elevation anomaly residual" of the GNSS leveling points calculated in the previous step, to obtain an accuracy grid model consistent with the resolution of the geoid model, is as follows: The coordinates of a certain grid point G in a geoid model are as follows: GNSS leveling points around it Coordinates are Then the precision of grid point G is: In the formula, The number of GNSS leveling points. Let be the "elevation anomaly residual" of the i-th GNSS leveling point; The weight of the i-th GNSS leveling point is defined as follows: in, For grid point G and GNSS leveling point distance, This is the equal-weighted radius; that is, when the distance is less than this value, this value is used instead of the actual distance. The search radius is [value].

[0047] In summary, the proposed strategy and method for generating quasi-geoid accuracy grid models can provide a method for calculating the accuracy of specific local models. This overcomes the problem that previous quasi-geoid model accuracy evaluations were not refined enough, only providing an overall model accuracy and failing to provide the accuracy of specific local models, thus failing to meet the needs of engineering applications.

[0048] Example 2 Using the "cyclic empty point method" to Figure 3 The test area for GNSS leveling points is shown. The check residuals of the GNSS leveling points in the test area are calculated. Based on the check residuals, the accuracy of the grid points is calculated. The geoid accuracy grid contour map of the test area is shown below. Figure 4 As shown. The distribution of grid points with high precision on the quasi-geoid surface is statistically analyzed, and the results are shown in [the table]. Figure 5As shown in Table 1, the accuracy is better in the northeastern part of the test area and slightly lower in the southern part. The format of the geoid accuracy grid model for the test area is the same as the grid model described earlier, stored in text. In the application, technicians provide the coordinates (longitude and latitude) of a certain location within the effective area of ​​the grid, and use the bilinear interpolation algorithm described above to interpolate the accuracy of the elevation anomaly value of the target location based on the geoid accuracy grid model.

[0049] Table 1. Statistical Table of Geoid Accuracy Grid Point Values ​​in the Western Experimental Area project Maximum value Minimum value mean Value / cm 16.5 2.6 8.5 The above description, in conjunction with specific preferred embodiments, provides a further detailed explanation of the present invention. It should not be construed that the specific implementation of the present invention is limited to these descriptions. For those skilled in the art, various simple deductions or substitutions can be made without departing from the concept of the present invention, and all such modifications and substitutions should be considered within the scope of protection of the present invention.

Claims

1. A strategy and method for generating a geoid-like grid model, characterized in that, Includes the following steps: S1. Using the "cyclic empty point method", the difference between the "measured elevation anomaly value" of each GNSS level point and the "model elevation anomaly value" obtained by interpolation based on the geoid model is calculated, which is the "elevation anomaly residual" of the GNSS level point. S2. Visit each grid point of the gravity-like geoid model one by one, search for GNSS leveling points within a certain range around it, and calculate the distance from the current grid point to each GNSS leveling point found. The search range is determined according to the distribution of GNSS leveling points around the current grid point, and at least 8 GNSS leveling points are required to fall within the search range. S3. Based on the distances between these GNSS leveling points and the current grid point, and the "elevation anomaly residuals" of the GNSS leveling points calculated in the previous step, calculate the accuracy of the current grid point to obtain an accuracy grid model consistent with the resolution of the geoid model.

2. The strategy and method for generating a geoid-like grid model as described in claim 1, characterized in that: S1, using the "cyclic empty point method," calculates the difference between the "measured elevation anomaly" of each GNSS leveling point and the "model elevation anomaly" obtained by interpolation based on the geoid model, i.e., the "elevation anomaly residual" of the GNSS leveling point. This specifically includes the following steps: S21. Among N GNSS leveling points, one point is reserved as a check point each time, and the other N-1 points are used as fitting points. S22. Subtract the normal height from the current check point to obtain its "measured elevation anomaly value"; in, Let be the "measured elevation anomaly" of the i-th GNSS leveling point. Let be the geodetic height of the i-th GNSS leveling point. This represents the normal elevation of the i-th GNSS leveling point; S23. Correct the gravity-based geoid model using the fitted points to obtain a fitted geoid model. The fitting steps are as follows: S231. Using bilinear interpolation, calculate the elevation anomaly values ​​of the fitted points based on the gravity-based geoid model. ; S232. Calculate the elevation outlier correction for each fitted point. ,Right now - ; S233. Visit each grid point in the gravity-based geoid model one by one, and search for at least 8 fitting points around it. Based on the elevation anomaly corrections of these fitting points, use the inverse distance weighting method to calculate the elevation anomaly correction of the grid point. Add this correction to the grid point value to obtain the fitted elevation anomaly of the grid point. After all grid points have been calculated, the fitted geoid model is obtained. Inverse distance weighting method: The coordinates of a certain grid point G in a geoid model are as follows: GNSS leveling points around it Coordinates are Then the elevation anomaly correction for grid point G is: In the formula, The number of GNSS leveling points. Let be the "elevation anomaly correction" for the i-th GNSS leveling point; The weight of the i-th GNSS leveling point is defined as follows: in, For grid point G and GNSS leveling point distance, This is the equal-weighted radius; that is, when the distance is less than this value, this value is used instead of the actual distance. The search radius; S24. Use bilinear interpolation to calculate the elevation anomaly value of the current check point in the fitted geoid grid model, i.e., the "model elevation anomaly value"; S25. Calculate the difference between the "measured elevation anomaly" and the "model elevation anomaly" at the current check point, i.e., the "elevation anomaly residual"; repeat the previous process to calculate the "elevation anomaly residual" for each GNSS leveling point. Let be the "elevation anomaly residual" of the i-th GNSS leveling point. Let be the "model elevation anomaly" of the i-th GNSS leveling point.

3. The strategy and method for generating a geoid-like grid model as described in claim 1, characterized in that: S2 involves sequentially visiting each grid point of the gravity-like geoid model, searching for GNSS leveling points within a certain radius of the current grid point, and calculating the distance from the current grid point to each of the searched GNSS leveling points. The distance calculation formula is as follows: In the formula, B1 and L1 are the latitude and longitude of the grid points, in °; B2 and L2 are the latitude and longitude of the GNSS leveling points, in °; and S is the distance, in km.

4. The strategy and method for generating a geoid-like grid model as described in claim 1, characterized in that: The specific process of S3, which involves calculating the accuracy of the current grid point based on the distances between these GNSS leveling points and the current grid point, and the "elevation anomaly residual" of the GNSS leveling points calculated in the previous step, to obtain an accuracy grid model consistent with the resolution of the geoid model, is as follows: The coordinates of a certain grid point G in a geoid model are as follows: GNSS leveling points around it Coordinates are Then the precision of grid point G is: In the formula, The number of GNSS leveling points. Let be the "elevation anomaly residual" of the i-th GNSS leveling point; The weight of the i-th GNSS leveling point is defined as follows: in, For grid point G and GNSS leveling point distance, This is the equal-weighted radius; that is, when the distance is less than this value, this value is used instead of the actual distance. The search radius is denoted as ; this method is called the inverse distance weighted method.

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